An Atlas of Matrices
Examples suitable for working by hand
Dave Bayer
c 2006, 2007 by Dave Bayer
All rights reserved
Projective Press
New York, NY USA
http://projectivepress.com/LinearAlgebra
Dave Bayer
Department of Mathematics
Barnard College
Columbia University
New York, New York
Draft of December 25, 2006
Contents
Contents
i
Preface
1
iii
Functions of matrices
1.1 2 × 2 matrices with distinct eigenvalues .
1.2 2 × 2 singular matrices . . . . . . . . . . .
1.3 2 × 2 symmetric matrices . . . . . . . . . .
1.4 2 × 2 matrices with a repeated eigenvalue
1.5 3 × 3 matrices with distinct eigenvalues .
1.6 3 × 3 singular matrices . . . . . . . . . . .
1.7 3 × 3 symmetric matrices . . . . . . . . . .
i
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1
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1
. 12
. 23
. 34
. 45
. 57
. 69
Preface
This is an early draft of examples in Linear Algebra, to accompany the text Linear Algebra. I am
making these drafts available in incomplete form for the benefit of my linear algebra students
at Barnard College and Columbia University.
This is a draft of December 25, 2006. The most recent drafts of these texts may be downloaded from:
http://projectivepress.com/LinearAlgebra
iii
Chapter 1
Functions of matrices
1.1
2 × 2 matrices with distinct eigenvalues
For each matrix below, the second form can be used to compute functions of matrices, such as
the matrix exponential. For example, if
1
2
−1
2
−1
0
−3
2
A =
=
3
2
1
3
0
4
1
1 /5
then
A
=
An
=
e At
=
3 −2
−1
−3
3
3 −2
(−1)n
−3
3
3 −2
e−t
−3
3
/5
/5
/5
2
3
+ 4n
2
3
+ e4t
2
3
+
4
2
3 /5
2
3 /5
2
3 /5
The following 2 × 2 matrices have nonzero integer entries, are nonsingular, have distinct
integer eigenvalues, and are not symmetric:
1
1
−1
1
−2
0
−1
1
=
3 −1
3
1
0
2
3
1 /4
1 −1
3
1
= −2
+ 2
−3
3 /4
3
1 /4
1
3
2
2
=
=
2
2
1
3
=
=
−1
1
0
−3
2
4
1
1 /5
3 −2
2
2
−1
+ 4
−3
2 /5
3
3 /5
−1
1
2
3
−1
0
1
0
0
−2
1
4
1
1 /3
2 −1
1
1
1
+ 4
−2
1 /3
2
2 /3
1
2
2
Functions of matrices
2
3
2
1
=
=
−1
3
1
1
=
=
3
2
1
2
=
=
−1
2
3 −2
=
=
3
2
3 −2
=
=
−2 −2
2
3
=
=
−2
2
3 −1
=
=
−2
3
2
3
=
=
−2 −1
3
2
=
=
−2
3
0
−1
1
4
3
2 /5
3
2
2 −2
+ 4
−1
3
2 /5
−3
3 /5
−1
1
−1
2
−2
3
−1
3
−2 −1
1
2
−1
1
−2
1
−1 −1
1
3
1
1
−1
0
−2
0
1
0
−4
0
−3
0
−1
0
−4
0
−3
0
−1
0
0
−3
1
2
1
1 /4
3 −1
1
1
−2
+ 2
−3
1 /4
3
3 /4
1
3
0
−1
1
4
2
1 /3
1 −1
2
1
1
+ 4
−2
2 /3
2
1 /3
1
1
0
−1
1
1
3
2 /5
2 −2
3
2
−4
+ 1
−3
3 /5
3
2 /5
1
1
0
−1
2
4
3
1 /7
1 −2
6
2
−3
+ 4
−3
6 /7
3
1 /7
2
1
0
−2 −1
2
1
2 /3
4
2
−1 −2
−1
+ 2
−2 −1 /3
2
4 /3
0
−3
2
1
1
1 /5
3 −2
2
2
−4
+ 1
−3
2 /5
3
3 /5
2
3
0
−3
1
4
1
2 /7
6 −2
1
2
−3
+ 4
3
6 /7
−3
1 /7
1
3
0
−3 −1
1
1
1 /2
3
1
−1 −1
−1
+ 1
−3 −1 /2
3
3 /2
1.1. 2 × 2 matrices with distinct eigenvalues
1 −2
1
4
=
=
1 −1
2
4
=
=
1
4
1
1
=
=
1
1
4 −2
=
=
1
2
4 −1
=
=
1
4
2
3
=
=
1
4
3
2
=
=
2
3
1
4
=
=
2
1
4 −1
=
=
3
−2 −1
1
1
2
0
−1 −1
1
2
2
0
−1
2
−1
0
−1
4
−3
0
−1
2
−3
0
−1
1
−1
0
−1
1
−2
0
−1
1
1
0
−1
4
−2
0
−1 −1
1
2
2
2
−1 −2
2
+ 3
−1 −1
1
2
0
3
−2 −1
1
1
2
1
−1 −1
2
+ 3
−2 −1
2
2
0
3
0
−2
1
3
2
1 /4
2 −1
2
1
−1
+ 3
−4
2 /4
4
2 /4
1
2
0
−1
1
2
4
1 /5
1 −1
4
1
−3
+ 2
−4
4 /5
4
1 /5
1
1
0
−1
1
3
2
1 /3
1 −1
2
1
−3
+ 3
−2
2 /3
2
1 /3
1
1
0
−2
1
5
1
1 /3
2 −1
1
1
−1
+ 5
−2
1 /3
2
2 /3
1
2
0
−4
3
5
1
1 /7
4 −3
3
3
−2
+ 5
−4
3 /7
4
4 /7
3
4
0
−3
1
5
1
1 /4
3 −1
1
1
1
+ 5
3
3 /4
−3
1 /4
1
3
0
−1
1
3
4
1 /5
1 −1
4
1
−2
+ 3
−4
4 /5
4
1 /5
1
1
4
Functions of matrices
2
4
3
1
=
=
2
4
3
3
=
=
2
3
4 −2
=
=
−1
3
2
4
=
=
−1 −2
3
4
=
=
−1
4
1
2
=
=
−1
1
4 −1
=
=
−1
4
2
1
=
=
−1
4
3
3
=
=
−3
4
−2
0
0
−1
1
5
4
3 /7
4
3
3 −3
+ 5
−2
4
3 /7
−4
4 /7
−1
1
−1
0
−1
2
−4
0
−2
1
1
3
−2
0
−1 −2
1
3
1
0
−1
1
−2
0
−1
2
−3
0
−1
1
−3
0
−3
2
−3
0
1
1
0
−4
3
6
1
1 /7
4 −3
3
3
−1
+ 6
−4
3 /7
4
4 /7
3
4
0
−2
3
4
2
1 /8
2 −3
6
3
−4
+ 4
−4
6 /8
4
2 /8
3
2
0
−3
1
5
1
2 /7
6 −2
1
2
−2
+ 5
−3
1 /7
3
6 /7
−3 −2
1
1
3
2
−2 −2
1
+ 2
−3 −2
3
3
0
2
0
−4
1
3
1
1 /5
4 −1
1
1
−2
+ 3
−4
1 /5
4
4 /5
1
4
0
−2
1
1
2
1 /4
2 −1
2
1
−3
+ 1
−4
2 /4
4
2 /4
1
2
0
−2
1
3
1
1 /3
2 −1
1
1
−3
+ 3
2
2 /3
−2
1 /3
1
2
0
−2
1
5
2
3 /8
6 −3
2
3
−3
+ 5
−4
2 /8
4
6 /8
1
2
1.1. 2 × 2 matrices with distinct eigenvalues
−1
3
4 −2
=
=
3
2
1
4
=
=
3
3
2
4
=
=
3
4
1
3
=
=
3
4
2
1
=
=
3
4
3
2
=
=
3
3
4 −1
=
=
−2
4
1
1
=
=
−2 −1
4
3
=
=
5
−3
4
0
−1
1
2
4
3 /7
4
3
3 −3
+ 2
−5
4
3 /7
−4
4 /7
−1
1
−1
1
−1
2
−1
2
−3
4
−1
2
−1
1
−1 −1
1
4
1
1
−5
0
2
0
1
0
1
0
−1
0
−1
0
−3
0
−3
0
−1
0
0
−2
1
5
1
1 /3
2 −1
1
1
2
+ 5
−2
1 /3
2
2 /3
1
2
0
−3
2
6
1
1 /5
3 −2
2
2
1
+ 6
−3
2 /5
3
3 /5
2
3
0
−2
1
5
2
1 /4
2 −1
2
1
1
+ 5
−4
2 /4
4
2 /4
1
2
0
−1
1
5
2
1 /3
1 −1
2
1
−1
+ 5
−2
2 /3
2
1 /3
1
1
0
−1
1
6
4
3 /7
3 −3
4
3
−1
+ 6
−4
4 /7
4
3 /7
1
1
0
−2
3
5
2
1 /8
2 −3
6
3
−3
+ 5
−4
6 /8
4
2 /8
3
2
0
−4
1
2
1
1 /5
4 −1
1
1
−3
+ 2
4
4 /5
−4
1 /5
1
4
0
−4 −1
2
1
1 /3
4
1
−1 −1
−1
+ 2
−4 −1 /3
4
4 /3
6
Functions of matrices
−2
4
3
2
=
=
−2
3
4 −1
=
=
4 −2
1
1
4
2
1
3
4 −1
2
1
=
4
3
1
2
4
2
3 −1
=
=
=
4
3
2
3
=
=
4
4
1
4
−1
1
1
1
−1
−1
−1
2
1
2
−1
−2
−1
3
−1
3
−2
3
−1
2
2
=
=
−4
0
3
4
−5
0
2
0
2
0
2
0
1
0
−2
0
1
0
2
0
−1
2
1 −1
2
2 −2
+ 3
2
1 −1
2
1
0
3
0
−1
1
5
2
1 /3
1 −1
2
1
2
+ 5
−2
2 /3
2
1 /3
=
=
1
2
0
−4
3
2
1
1 /7
4 −3
3
3
−5
+ 2
−4
3 /7
4
4 /7
=
−3
2
=
=
0
−2
1
4
2
3 /8
2
3
6 −3
+ 4
−4
4
6 /8
−4
2 /8
=
2
1
1
−1
1
2 −1
1
2 −1
+ 3
2
2 −1
1
1
0
3
0
−1
1
5
3
1 /4
1 −1
3
1
1
+ 5
−3
3 /4
3
1 /4
1
1
0
−1
2
5
3
1 /7
1 −2
6
2
−2
+ 5
−3
6 /7
3
1 /7
2
1
0
−1
1
6
3
2 /5
2 −2
3
2
1
+ 6
3
2 /5
−3
3 /5
1
1
0
−2
1
6
2
1 /4
2 −1
2
1
2
+ 6
−4
2 /4
4
2 /4
1
2
1.1. 2 × 2 matrices with distinct eigenvalues
1
3
4 −3
=
=
2
2
3 −3
=
=
−1 −3
2
4
=
=
−1
2
4 −3
=
=
−2 −3
1
2
=
=
−2
1
2 −3
−2
3
4 −3
4
2
4 −3
−3
1
2 −2
−5
0
0
−2
3
3
2
1 /8
6
3
2 −3
+ 3
−5
4
2 /8
−4
6 /8
−1
3
2
1
−4
0
−3 −1
2
1
1
0
−1
2
−5
0
−3 −1
1
1
−1
0
−1
2
−3
4
−1
4
=
=
−1
1
3
2
0
−1
2
3
3
1 /7
1 −2
6
2
−4
+ 3
−3
6 /7
3
1 /7
−1 −1
2
3
3
3
−2 −3
1
+ 2
−2 −2
2
3
0
2
0
−1
1
1
2
1 /3
1 −1
2
1
−5
+ 1
−2
2 /3
2
1 /3
1
1
0
−1 −1
1
1
3 /2
3
3
−1 −3
−1
+ 1
−1 −1 /2
1
3 /2
−4
0
−1
1
0 −1
2
1 /3
1 −1
2
1
= −4
− 1
−2
2 /3
2
1 /3
=
−1
2
=
=
7
1
1
0
−1
1
1
4
3 /7
3 −3
4
3
−6
+ 1
−4
4 /7
4
3 /7
1
1
−6
0
−4
0
0
−1
2
5
4
1 /9
1 −2
8
2
−4
+ 5
4
1 /9
−4
8 /9
2
1
−4
0
−2
1
=
0 −1
1
1 /3
2 −1
1
1
= −4
− 1
−2
1 /3
2
2 /3
1
2
8
Functions of matrices
−3 −2
2
2
=
=
−3 −3
2
4
=
=
−3
3
2
2
=
=
−3 −2
3
4
=
=
−3
1
4 −3
−3
2
4 −1
−3
4
2
4
=
=
=
−3 −1
4
2
=
=
−3
4
3
1
−2 −1
1
2
−3 −1
1
2
−2
1
−2 −1
1
3
−1
2
−1
1
−2
1
−1 −1
1
4
−3
2
−2
0
0
−2 −1
1
1
2 /3
−1 −2
4
2
+ 1
−2
2
4 /3
−2 −1 /3
−2
0
−4
0
−2
0
0
−2 −1
3
1
3 /5
6
3
−1 −3
−2
+ 3
−2 −1 /5
2
6 /5
0
−3
1
3
1
2 /7
6 −2
1
2
−4
+ 3
−3
1 /7
3
6 /7
1
3
0
−3 −1
3
1
2 /5
6
2
−1 −2
−2
+ 3
−3 −1 /5
3
6 /5
−5
0
−2
1
=
0 −1
2
1 /4
2 −1
2
1
= −5
− 1
−4
2 /4
4
2 /4
=
=
=
1
2
0
−2
1
1
1
1 /3
2 −1
1
1
−5
+ 1
−2
1 /3
2
2 /3
1
2
−5
0
−4
0
−2
0
−5
0
0
−4
1
5
1
2 /9
8 −2
1
2
−4
+ 5
−4
1 /9
4
8 /9
1
4
0
−4 −1
1
1
1 /3
4
1
−1 −1
−2
+ 1
4
4 /3
−4 −1 /3
0
−2
1
3
2
3 /8
6 −3
2
3
−5
+ 3
−4
2 /8
4
6 /8
1
2
1.1. 2 × 2 matrices with distinct eigenvalues
−3
3
4 −2
=
=
−3 −2
4
3
1 −3
1
5
1 −1
3
5
=
=
=
1
4
3
5
=
=
1
1
5 −3
=
=
1
2
5 −2
=
=
1
5
2
4
=
=
1
3
5 −1
−1
1
3
4
−6
0
0
−4
3
1
1
1 /7
3
3
4 −3
+ 1
−6
4
4 /7
−4
3 /7
−1 −1
1
2
−1
0
−3 −1
1
1
2
0
−1 −1
1
3
2
0
−3
2
−1
0
−1
5
−4
0
−2
5
−4
0
−1
1
−1
0
−3
5
−4
0
−2 −1
=
1
1
2
1
−1 −1
= −1
+ 1
−2 −1
2
2
=
9
=
=
0
1
0
−1 −1
4
1
3 /2
3
3
−1 −3
2
+ 4
−1 −1 /2
1
3 /2
0
−3 −1
4
1
1 /2
3
1
−1 −1
2
+ 4
−3 −1 /2
3
3 /2
0
−2
1
7
2
3 /8
6 −3
2
3
−1
+ 7
−4
2 /8
4
6 /8
1
2
0
−1
1
2
5
1 /6
1 −1
5
1
−4
+ 2
−5
5 /6
5
1 /6
1
1
0
−1
1
3
5
2 /7
2 −2
5
2
−4
+ 3
−5
5 /7
5
2 /7
1
1
0
−5
2
6
1
1 /7
5 −2
2
2
−1
+ 6
5
5 /7
−5
2 /7
2
5
0
−1
1
4
5
3 /8
3 −3
5
3
−4
+ 4
−5
5 /8
5
3 /8
1
1
10
Functions of matrices
1
5
3
3
=
=
1
5
4
2
=
=
2 −2
1
5
=
=
2 −1
2
5
=
=
2
4
1
5
=
=
2
1
5 −2
=
=
2
2
5 −1
=
=
2
5
3
4
=
=
2
5
4
1
=
=
−1
1
−2
0
0
−5
3
6
1
1 /8
3
3
5 −3
+ 6
−2
5
5 /8
−5
3 /8
−1
1
4
5
−3
0
−2 −1
1
1
3
0
−1 −1
1
2
3
0
−1
1
1
0
−1
5
−3
0
−2
5
−3
0
−1
1
−1
0
−4
5
−3
0
3
5
0
−5
4
6
1
1 /9
5 −4
4
4
−3
+ 6
−5
4 /9
5
5 /9
−1 −1
1
2
2
2
−1 −2
3
+ 4
−1 −1
1
2
0
4
−2 −1
1
1
2
1
−1 −1
3
+ 4
−2 −1
2
2
0
4
0
−4
1
6
1
1 /5
4 −1
1
1
1
+ 6
−4
1 /5
4
4 /5
1
4
0
−1
1
3
5
1 /6
1 −1
5
1
−3
+ 3
−5
5 /6
5
1 /6
1
1
0
−1
1
4
5
2 /7
2 −2
5
2
−3
+ 4
−5
5 /7
5
2 /7
1
1
0
−5
3
7
1
1 /8
5 −3
3
3
−1
+ 7
5
5 /8
−5
3 /8
3
5
0
−1
1
6
5
4 /9
4 −4
5
4
−3
+ 6
−5
5 /9
5
4 /9
1
1
1.1. 2 × 2 matrices with distinct eigenvalues
2
5
4
3
=
=
−1 −2
4
5
=
=
−1
5
1
3
=
=
−1
5
2
2
=
=
−1
5
3
1
=
=
−1
3
5 −3
=
=
−1
4
5 −2
=
=
11
−1
1
4
5
−2
0
0
−5
4
7
1
1 /9
4
4
5 −4
+ 7
−2
5
5 /9
−5
4 /9
−1 −1
1
2
1
0
−1
1
−2
0
−1
1
−3
0
−1
1
−4
0
−3
5
−6
0
−4
5
−6
0
−2 −1
1
1
2
1
−1 −1
1
+ 3
−2 −1
2
2
0
3
0
−5
1
4
1
1 /6
5 −1
1
1
−2
+ 4
−5
1 /6
5
5 /6
1
5
0
−5
2
4
1
1 /7
5 −2
2
2
−3
+ 4
−5
2 /7
5
5 /7
2
5
0
−5
3
4
1
1 /8
5 −3
3
3
−4
+ 4
−5
3 /8
5
5 /8
3
5
0
−1
1
2
5
3 /8
3 −3
5
3
−6
+ 2
−5
5 /8
5
3 /8
1
1
0
−1
1
3
5
4 /9
4 −4
5
4
−6
+ 3
−5
5 /9
5
4 /9
1
1
12
1.2
Functions of matrices
2 × 2 singular matrices
For each matrix below, the second form can be used to compute functions of matrices, such as
the matrix exponential. For example, if
2
1
−1
1
0
0
−1
1
A =
=
2
1
2
1
0
3
2
1 /3
then
A
=
An
=
e At
=
1 −1
2
0
+ 3
−2
2 /3
2
1 −1
2
0n
+ 3n
−2
2 /3
2
1 −1
2
e0t
+ e3t
−2
2 /3
2
1
1 /3
1
1 /3
1
1 /3
The following 2 × 2 matrices have nonzero integer entries, are singular, have distinct integer
eigenvalues, and are not symmetric:
1
1
−1
1
0
0
−2
1
=
2
2
1
2
0
3
1
1 /3
2 −1
1
1
=
0
+ 3
−2
1 /3
2
2 /3
2
2
1
1
=
=
−1 −1
2
2
=
=
1
3
1
3
=
=
2
3
2
3
=
=
−1
2
1
1
0
0
−1 −1
1
2
0
0
−1
1
0
0
−1
1
0
0
0
−1
1
3
2
1 /3
1 −1
2
1
0
+ 3
−2
2 /3
2
1 /3
−2 −1
1
1
2
1
−1 −1
0
+ 1
−2 −1
2
2
0
1
0
−3
1
4
1
1 /4
3 −1
1
1
0
+ 4
−3
1 /4
3
3 /4
1
3
0
−3
2
5
1
1 /5
3 −2
2
2
0
+ 5
−3
2 /5
3
3 /5
2
3
1.2. 2 × 2 singular matrices
−1 −1
3
3
13
=
=
3
3
1
1
=
=
3
3
2
2
=
=
−1 −2
1
2
=
=
−1
1
2 −2
=
=
−2 −2
1
1
−2
1
2 −1
−1 −1
1
3
0
0
0
−3 −1
2
1
1 /2
−1 −1
3
1
+ 2
0
3
3 /2
−3 −1 /2
−1
3
0
0
−2
3
1
1
0
0
−2 −1
1
1
0
0
−1
2
1
1
−3
0
−2 −1
1
1
−1
0
−1
1
1
2
−3
0
−1 −1
1
2
−1
0
−1 −2
1
3
0
0
0
−1
1
4
3
1 /4
1 −1
3
1
0
+ 4
−3
3 /4
3
1 /4
−2 −2
3
3
−1 −1
1
2
2
2
−1 −2
0
+ 1
−1 −1
1
2
0
1
0
0
−2 −1
1
1
2
1
−1 −1
= −1
+ 0
−2 −1
2
2
=
0
−2
1
0
1
1 /3
2 −1
1
1
−3
+ 0
−2
1 /3
2
2 /3
=
=
0
−1
1
0
2
1 /3
1 −1
2
1
−3
+ 0
−2
2 /3
2
1 /3
−1 −1
1
2
2
2
−1 −2
= −1
+ 0
−1 −1
1
2
=
−2 −1
2
1
1
1
0
−1
1
5
3
2 /5
2 −2
3
2
0
+ 5
−3
3 /5
3
2 /5
=
=
0
0
−3 −2
1
1
3
2
−2 −2
0
+ 1
−3 −2
3
3
0
1
14
Functions of matrices
1
4
1
4
=
=
2
4
1
2
=
=
2
4
2
4
=
=
−1 −2
2
4
=
=
−1 −1
4
4
=
=
3
4
3
4
=
=
−2
1
4 −2
=
=
−2 −2
4
4
=
=
4
4
1
1
=
=
−1
1
0
0
0
−4
1
5
1
1 /5
1
1
4 −1
+ 5
0
4
4 /5
−4
1 /5
−1
2
0
0
−1
1
0
0
−2 −1
1
2
0
0
−1 −1
1
4
0
0
−1
1
0
0
−1
2
1
2
−4
0
−1 −1
1
2
0
0
−1
4
0
0
1
4
0
−2
1
4
2
1 /4
2 −1
2
1
0
+ 4
−4
2 /4
4
2 /4
1
2
0
−2
1
6
1
1 /3
2 −1
1
1
0
+ 6
−2
1 /3
2
2 /3
1
2
0
−2 −1
3
1
2 /3
4
2
−1 −2
0
+ 3
−2 −1 /3
2
4 /3
0
−4 −1
3
1
1 /3
4
1
−1 −1
0
+ 3
−4 −1 /3
4
4 /3
0
−4
3
7
1
1 /7
4 −3
3
3
0
+ 7
−4
3 /7
4
4 /7
3
4
0
−2
1
0
2
1 /4
2 −1
2
1
−4
+ 0
−4
2 /4
4
2 /4
−2 −1
1
1
2
1
−1 −1
0
+ 2
−2 −1
2
2
0
2
0
−1
1
5
4
1 /5
1 −1
4
1
0
+ 5
−4
4 /5
4
1 /5
1
1
1.2. 2 × 2 singular matrices
4
4
2
2
15
=
=
4
4
3
3
=
=
−1 −3
1
3
=
=
−1
1
3 −3
=
=
−2 −3
2
3
=
=
−2
2
3 −3
=
=
−3 −3
1
1
=
=
−3 −3
2
2
−3
1
3 −1
−1
2
0
0
0
−1
1
6
2
1 /3
2
1
1 −1
+ 6
0
2
1 /3
−2
2 /3
−3
4
0
0
−3 −1
1
1
0
0
−1
3
1
1
−4
0
−3 −1
2
1
0
0
−2
3
−5
0
−3 −1
1
1
−2
0
−3 −1
2
1
−1
0
−1
1
−4
0
1
1
0
−1
1
7
4
3 /7
3 −3
4
3
0
+ 7
−4
4 /7
4
3 /7
1
1
0
−1 −1
2
1
3 /2
3
3
−1 −3
0
+ 2
−1 −1 /2
1
3 /2
0
−1
1
0
3
1 /4
1 −1
3
1
−4
+ 0
−3
3 /4
3
1 /4
−1 −1
2
3
3
3
−2 −3
0
+ 1
−2 −2
2
3
0
1
0
−1
1
0
3
2 /5
2 −2
3
2
−5
+ 0
−3
3 /5
3
2 /5
1
1
0
−1 −1
0
1
3 /2
3
3
−1 −3
−2
+ 0
−1 −1 /2
1
3 /2
−1 −1
2
3
3
3
−2 −3
= −1
+ 0
−2 −2
2
3
=
=
=
0
0
0
−3
1
0
1
1 /4
3 −1
1
1
−4
+ 0
−3
1 /4
3
3 /4
1
3
16
Functions of matrices
−3
2
3 −2
=
=
−3 −1
3
1
=
=
−3 −2
3
2
−3 −3
4
4
1
5
1
5
2
5
2
5
=
=
=
=
−1 −1
5
5
=
=
3
5
3
5
=
=
−2 −2
5
5
−5
0
0
−3
2
0
1
1 /5
2
2
3 −2
+ 0
−5
3
3 /5
−3
2 /5
−1 −1
1
3
−2
0
−1 −2
1
3
−1
0
−1 −3
1
4
0
0
−1
1
0
0
−1
1
0
0
−1 −1
1
5
0
0
−1
1
0
0
−1 −2
1
5
0
0
2
3
0
−3 −1
0
1
1 /2
3
1
−1 −1
−2
+ 0
−3 −1 /2
3
3 /2
−3 −2
1
1
3
2
−2 −2
= −1
+ 0
−3 −2
3
3
=
−1
1
=
=
=
=
0
0
−4 −3
1
1
4
3
−3 −3
0
+ 1
−4 −3
4
4
0
1
0
−5
1
6
1
1 /6
5 −1
1
1
0
+ 6
−5
1 /6
5
5 /6
1
5
0
−5
2
7
1
1 /7
5 −2
2
2
0
+ 7
−5
2 /7
5
5 /7
2
5
0
−5 −1
4
1
1 /4
5
1
−1 −1
0
+ 4
−5 −1 /4
5
5 /4
0
−5
3
8
1
1 /8
5 −3
3
3
0
+ 8
5
5 /8
−5
3 /8
3
5
0
−5 −2
3
1
1 /3
5
2
−2 −2
0
+ 3
−5 −2 /3
5
5 /3
1.2. 2 × 2 singular matrices
4
5
4
5
17
=
=
−3 −3
5
5
=
=
5
5
1
1
=
=
5
5
2
2
=
=
5
5
3
3
=
=
5
5
4
4
=
=
−1 −4
1
4
=
=
−1
1
4 −4
=
=
−2 −4
2
4
=
=
−1
1
0
0
0
−5
4
9
1
1 /9
4
4
5 −4
+ 9
0
5
5 /9
−5
4 /9
−1 −3
1
5
0
0
−1
5
0
0
−2
5
0
0
−3
5
0
0
−4
5
0
0
−4 −1
1
1
0
0
−1
4
1
1
−5
0
−2 −1
1
1
0
0
4
5
0
−5 −3
2
1
1 /2
5
3
−3 −3
0
+ 2
−5 −3 /2
5
5 /2
0
−1
1
6
5
1 /6
1 −1
5
1
0
+ 6
−5
5 /6
5
1 /6
1
1
0
−1
1
7
5
2 /7
2 −2
5
2
0
+ 7
−5
5 /7
5
2 /7
1
1
0
−1
1
8
5
3 /8
3 −3
5
3
0
+ 8
−5
5 /8
5
3 /8
1
1
0
−1
1
9
5
4 /9
4 −4
5
4
0
+ 9
−5
5 /9
5
4 /9
1
1
0
−1 −1
3
1
4 /3
4
4
−1 −4
0
+ 3
−1 −1 /3
1
4 /3
0
−1
1
0
4
1 /5
1 −1
4
1
−5
+ 0
4
1 /5
−4
4 /5
−1 −1
1
2
2
2
−1 −2
0
+ 2
−1 −1
1
2
0
2
18
Functions of matrices
−2
2
4 −4
=
=
−3 −4
3
4
=
=
−3
3
4 −4
=
=
−4 −4
1
1
=
=
−4 −2
2
1
=
=
−4 −4
2
2
−4 −4
3
3
−4
1
4 −1
−4
2
4 −2
−1
2
1
1
−6
0
0
−1
1
0
2
1 /3
2
1
1 −1
+ 0
−6
2
1 /3
−2
2 /3
−4 −1
3
1
0
0
−3
4
−7
0
−4 −1
1
1
−3
0
−2 −1
1
2
−3
0
−2 −1
1
1
−2
0
0
0
−4 −1
3
1
−1
0
0
0
−1
1
−5
0
−1
1
−6
0
−1 −1
3
4
4
4
−3 −4
0
+ 1
−3 −3
3
4
0
1
1
1
0
−1 −1
0
1
4 /3
4
4
−1 −4
−3
+ 0
−1 −1 /3
1
4 /3
0
−2 −1
0
1
2 /3
4
2
−1 −2
−3
+ 0
−2 −1 /3
2
4 /3
−1 −1
1
2
2
2
−1 −2
= −2
+ 0
−1 −1
1
2
−1 −1
=
3
4
4
4
−3 −4
= −1
+ 0
−3 −3
3
4
=
=
=
0
−1
1
0
4
3 /7
3 −3
4
3
−7
+ 0
−4
4 /7
4
3 /7
=
=
0
−4
1
0
1
1 /5
4 −1
1
1
−5
+ 0
−4
1 /5
4
4 /5
1
4
0
−2
1
0
1
1 /3
2 −1
1
1
−6
+ 0
−2
1 /3
2
2 /3
1
2
1.2. 2 × 2 singular matrices
−4 −1
4
1
19
=
=
−4
3
4 −3
=
=
−4 −2
4
2
−4 −3
4
3
−4 −4
5
5
1
3
2
6
1
6
1
6
2
4
3
6
2
6
1
3
0
−4 −1
0
1
1 /3
−1 −1
4
1
+ 0
−3
4
4 /3
−4 −1 /3
−1
1
3
4
−7
0
−1 −1
1
2
−2
0
0
0
−1 −3
1
4
−1
0
0
0
−1 −4
1
5
0
0
−2
1
0
0
−1
1
0
0
−3
2
0
0
−1
2
0
0
0
−4
3
0
1
1 /7
4 −3
3
3
−7
+ 0
−4
3 /7
4
4 /7
−4 −3
1
1
4
3
−3 −3
= −1
+ 0
−4 −3
4
4
=
=
=
=
=
−3
0
=
=
−2 −1
1
1
2
1
−1 −1
= −2
+ 0
−2 −1
2
2
=
−1 −1
1
4
=
=
=
=
−5 −4
1
1
5
4
−4 −4
0
+ 1
−5 −4
5
5
0
1
0
−3
1
7
1
2 /7
6 −2
1
2
0
+ 7
−3
1 /7
3
6 /7
1
3
0
−6
1
7
1
1 /7
6 −1
1
1
0
+ 7
−6
1 /7
6
6 /7
1
6
0
−2
1
8
2
3 /8
6 −3
2
3
0
+ 8
−4
2 /8
4
6 /8
1
2
0
−3
1
5
2
1 /5
3 −1
2
1
0
+ 5
−6
2 /5
6
3 /5
1
3
20
Functions of matrices
2
6
2
6
=
=
−1 −3
2
6
=
=
−1 −2
3
6
=
=
−1 −1
6
6
=
=
3
6
1
2
=
=
3
6
2
4
=
=
3
6
3
6
=
=
−2 −4
3
6
=
=
−2 −3
4
6
=
=
−1
1
0
−3
1
8
1
1 /4
1
1
3 −1
+ 8
0
3
3 /4
−3
1 /4
−3 −1
1
2
−2 −1
1
3
−1 −1
1
6
−1
3
−2
3
−1
1
−2 −2
1
3
−3 −1
2
2
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
−2 −1
5
1
3 /5
6
3
−1 −3
0
+ 5
−2 −1 /5
2
6 /5
0
−3 −1
5
1
2 /5
6
2
−1 −2
0
+ 5
−3 −1 /5
3
6 /5
0
−6 −1
5
1
1 /5
6
1
−1 −1
0
+ 5
−6 −1 /5
6
6 /5
0
−2
1
5
3
1 /5
2 −1
3
1
0
+ 5
−6
3 /5
6
2 /5
1
2
0
−2
1
7
3
2 /7
4 −2
3
2
0
+ 7
−6
3 /7
6
4 /7
1
2
0
−2
1
9
1
1 /3
2 −1
1
1
0
+ 9
−2
1 /3
2
2 /3
1
2
0
−3 −2
4
1
2 /4
6
4
−2 −4
0
+ 4
3
6 /4
−3 −2 /4
0
−2 −1
4
2
3 /4
6
3
−2 −3
0
+ 4
−4 −2 /4
4
6 /4
1.2. 2 × 2 singular matrices
−2
1
6 −3
21
=
=
−2 −1
6
3
=
=
−2 −2
6
6
=
=
4
6
2
3
=
=
4
6
4
6
−3
1
6 −2
−3
2
6 −4
=
=
=
−1
3
1
2
−5
0
−1 −1
2
3
0
0
−1 −1
1
3
0
0
−1
2
2
3
0
0
−1
1
2
3
0
0
3 −2
2
+ 10
−3
2 /5
3
−1
2
−1
2
−3 −1
6
2
−3 −2
6
4
−3 −1
2
1
3
1
−2 −1
0
+ 1
−6 −2
6
3
0
1
0
−3
2
7
2
1 /7
3 −2
4
2
0
+ 7
−6
4 /7
6
3 /7
0
−5
0
2
3
−7
0
−1 −1
2
3
−1
0
−2 −1
3
2
0
0
0
10
−3
1
2
1 /5
2
3 /5
0
−3
1
0
2
1 /5
3 −1
2
1
−5
+ 0
−6
2 /5
6
3 /5
1
3
0
−3
2
0
2
1 /7
3 −2
4
2
−7
+ 0
−6
4 /7
6
3 /7
−3 −1
2
1
3
1
−2 −1
= −1
+ 0
−6 −2
6
3
=
=
=
0
−3 −1
4
1
1 /2
3
1
−1 −1
0
+ 4
−3 −1 /2
3
3 /2
=
=
0
−2
1
0
3
1 /5
3
1
2 −1
+ 0
−5
6
2 /5
−6
3 /5
=
0
0
−2 −1
3
2
4
2
−3 −2
0
+ 1
−6 −3
6
4
0
1
22
Functions of matrices
−3 −3
6
6
=
=
5
6
5
6
−1 −1
1
2
0
0
0
3
−1
1
0
0
0
11
−2 −1
1
1
2
1
−1 −1
0
+ 3
−2 −1
2
2
=
5
6
=
−4
2
6 −3
=
=
−4 −2
6
3
−4 −4
6
6
6
3
2
1
6
4
3
2
−2
3
1
2
−7
0
−2 −1
3
2
−1
0
−1 −2
1
3
0
0
−1
3
0
0
−1
2
0
0
5
1 /11
5
6 /11
0
−2
1
0
3
2 /7
4 −2
3
2
−7
+ 0
−6
3 /7
6
4 /7
−2 −1
3
2
4
2
−3 −2
= −1
+ 0
−6 −3
6
4
=
=
=
6 −5
5
0
+ 11
−6
5 /11
6
=
=
−6
1
=
=
0
0
−3 −2
1
1
3
2
−2 −2
0
+ 2
−3 −2
3
3
0
2
0
−1
2
7
3
1 /7
1 −2
6
2
0
+ 7
−3
6 /7
3
1 /7
2
1
0
−2
3
8
2
1 /8
2 −3
6
3
0
+ 8
−4
6 /8
4
2 /8
3
2
1.3. 2 × 2 symmetric matrices
1.3
23
2 × 2 symmetric matrices
For each matrix below, the second form can be used to compute functions of matrices, such as
the matrix exponential. For example, if
2
1
−1
1
1
0
−1
1
A =
=
1
2
1
1
0
3
1
1 /2
then
A
=
An
=
e At
=
1 −1
1
1
+ 3
−1
1 /2
1
1 −1
1
1n
+ 3n
−1
1 /2
1
1 −1
1
et
+ e3t
−1
1 /2
1
1
1 /2
1
1 /2
1
1 /2
The following 2 × 2 matrices have nonzero integer entries, are nonsingular, have distinct
integer eigenvalues, and are symmetric:
1
2
−1
1
−1
0
−1
1
=
2
1
1
1
0
3
1
1 /2
1 −1
1
1
= −1
+ 3
−1
1 /2
1
1 /2
2
1
1
2
=
=
2
2
2 −1
=
=
−1
2
2
2
=
=
−1
2
2 −1
=
=
−1
1
−1
2
−2
1
−1
1
1
0
−2
0
−2
0
−3
0
0
−1
1
3
1
1 /2
1 −1
1
1
1
+ 3
−1
1 /2
1
1 /2
1
1
0
−1
2
3
2
1 /5
1 −2
4
2
−2
+ 3
−2
4 /5
2
1 /5
2
1
0
−2
1
3
1
2 /5
4 −2
1
2
−2
+ 3
−2
1 /5
2
4 /5
1
2
0
−1
1
1
1
1 /2
1 −1
1
1
−3
+ 1
1
1 /2
−1
1 /2
1
1
24
Functions of matrices
1
3
3
1
=
=
2
3
3
2
=
=
−1
3
3 −1
=
=
3
1
1
3
=
=
3
2
2
3
=
=
1
2
2 −2
=
=
−2
1
1 −2
=
=
−2
2
2
1
=
=
−2
3
3 −2
=
=
−1
1
0
−1
1
4
1
1 /2
1
1
1 −1
+ 4
−2
1
1 /2
−1
1 /2
−1
1
−1
1
−1
1
−1
1
−1
2
−1
1
−2
1
−1
1
1
1
−2
0
−1
0
−4
0
2
0
1
0
−3
0
0
−1
1
5
1
1 /2
1 −1
1
1
−1
+ 5
−1
1 /2
1
1 /2
1
1
0
−1
1
2
1
1 /2
1 −1
1
1
−4
+ 2
−1
1 /2
1
1 /2
1
1
0
−1
1
4
1
1 /2
1 −1
1
1
2
+ 4
−1
1 /2
1
1 /2
1
1
0
−1
1
5
1
1 /2
1 −1
1
1
1
+ 5
−1
1 /2
1
1 /2
1
1
0
−1
2
2
2
1 /5
1 −2
4
2
−3
+ 2
−2
4 /5
2
1 /5
2
1
−3
0
−1
1
0 −1
1
1 /2
1 −1
1
1
−3
− 1
−1
1 /2
1
1 /2
1
1
0
−2
1
2
1
2 /5
4 −2
1
2
−3
+ 2
2
4 /5
−2
1 /5
1
2
−3
0
−5
0
0
−1
1
1
1
1 /2
1 −1
1
1
−5
+ 1
−1
1 /2
1
1 /2
1
1
1.3. 2 × 2 symmetric matrices
1
4
4
1
=
=
2
4
4
2
=
=
−1
4
4 −1
=
=
3
4
4
3
=
=
−2
4
4 −2
=
=
−2
4
4
4
=
=
4
1
1
4
=
=
4
2
2
4
=
=
4
3
3
4
=
=
25
−1
1
0
−1
1
5
1
1 /2
1
1
1 −1
+ 5
−3
1
1 /2
−1
1 /2
−1
1
−1
1
−1
1
−1
1
−2
1
−1
1
−1
1
−1
1
1
1
−3
0
−2
0
−5
0
−1
0
−6
0
−4
0
3
0
2
0
1
0
0
−1
1
6
1
1 /2
1 −1
1
1
−2
+ 6
−1
1 /2
1
1 /2
1
1
0
−1
1
3
1
1 /2
1 −1
1
1
−5
+ 3
−1
1 /2
1
1 /2
1
1
0
−1
1
7
1
1 /2
1 −1
1
1
−1
+ 7
−1
1 /2
1
1 /2
1
1
0
−1
1
2
1
1 /2
1 −1
1
1
−6
+ 2
−1
1 /2
1
1 /2
1
1
0
−2
1
6
1
2 /5
4 −2
1
2
−4
+ 6
−2
1 /5
2
4 /5
1
2
0
−1
1
5
1
1 /2
1 −1
1
1
3
+ 5
−1
1 /2
1
1 /2
1
1
0
−1
1
6
1
1 /2
1 −1
1
1
2
+ 6
1
1 /2
−1
1 /2
1
1
0
−1
1
7
1
1 /2
1 −1
1
1
1
+ 7
−1
1 /2
1
1 /2
1
1
26
Functions of matrices
4
4
4 −2
=
=
3
4
4 −3
=
=
−3
1
1 −3
=
=
−3
2
2 −3
=
=
−3
4
4
3
=
=
−3
4
4 −3
=
=
1
5
5
1
=
=
2
2
2
5
=
=
2
5
5
2
=
=
−1
2
0
−1
2
6
2
1 /5
4
2
1 −2
+ 6
−4
2
1 /5
−2
4 /5
−1
2
−1
1
−1
1
−2
1
−1
1
−1
1
−2
1
−1
1
2
1
−4
0
−5
0
0
−1
2
5
2
1 /5
1 −2
4
2
−5
+ 5
−2
4 /5
2
1 /5
2
1
−4
0
−1
1
0 −2
1
1 /2
1 −1
1
1
−4
− 2
−1
1 /2
1
1 /2
1
1
−5
0
−1
1
0 −1
1
1 /2
1 −1
1
1
−5
− 1
−1
1 /2
1
1 /2
1
1
0
−2
1
5
1
2 /5
4 −2
1
2
−5
+ 5
−2
1 /5
2
4 /5
1
2
−5
0
−7
0
−4
0
1
0
−3
0
0
−1
1
1
1
1 /2
1 −1
1
1
−7
+ 1
−1
1 /2
1
1 /2
1
1
0
−1
1
6
1
1 /2
1 −1
1
1
−4
+ 6
−1
1 /2
1
1 /2
1
1
0
−2
1
6
1
2 /5
4 −2
1
2
1
+ 6
2
4 /5
−2
1 /5
1
2
0
−1
1
7
1
1 /2
1 −1
1
1
−3
+ 7
−1
1 /2
1
1 /2
1
1
1.3. 2 × 2 symmetric matrices
−1
4
4
5
=
=
−1
5
5 −1
=
=
3
5
5
3
=
=
−2
5
5 −2
=
=
4
5
5
4
=
=
−3
3
3
5
27
−2
1
−1
1
−1
1
−1
1
−1
1
−3
1
−3
0
0
−2
1
7
1
2 /5
1
2
4 −2
+ 7
−3
2
4 /5
−2
1 /5
−6
0
−2
0
−7
0
1
1
−1
0
1
3
−4
0
1
2
0
−1
1
4
1
1 /2
1 −1
1
1
−6
+ 4
−1
1 /2
1
1 /2
1
1
0
−1
1
8
1
1 /2
1 −1
1
1
−2
+ 8
−1
1 /2
1
1 /2
1
1
0
−1
1
3
1
1 /2
1 −1
1
1
−7
+ 3
−1
1 /2
1
1 /2
1
1
0
−1
1
9
1
1 /2
1 −1
1
1
−1
+ 9
−1
1 /2
1
1 /2
=
=
−3
5
5 −3
=
=
5
1
1
5
=
=
5
2
2
2
=
=
0
6
−3
1
9 −3
1
−4
+ 6
−3
1 /10
3
−1
1
−1
1
−1
2
−8
0
4
0
1
0
3
9 /10
0
−1
1
2
1
1 /2
1 −1
1
1
−8
+ 2
−1
1 /2
1
1 /2
1
1
1
3 /10
0
−1
1
6
1
1 /2
1 −1
1
1
4
+ 6
1
1 /2
−1
1 /2
1
1
0
−1
2
6
2
1 /5
1 −2
4
2
1
+ 6
−2
4 /5
2
1 /5
2
1
28
Functions of matrices
5
2
2
5
=
=
5
3
3 −3
−1
1
1
1
3
0
0
−1
1
7
1
1 /2
1
1
1 −1
+ 7
3
1
1 /2
−1
1 /2
−1
3
3
1
−4
0
=
0
6
−1
3
=
5
3
3
5
=
=
5
4
4 −1
=
=
5
4
4
5
=
=
2
4
4 −4
=
=
4
3
3 −4
1 −3
9
−4
+ 6
−3
9 /10
3
−1
1
−1
2
−1
1
−1
2
−1
3
2
0
−3
0
1
0
2
1
−6
0
3
1
−5
0
1
1
0
−1
2
7
2
1 /5
1 −2
4
2
−3
+ 7
−2
4 /5
2
1 /5
2
1
0
−1
1
9
1
1 /2
1 −1
1
1
1
+ 9
−1
1 /2
1
1 /2
1
1
0
−1
2
4
2
1 /5
1 −2
4
2
−6
+ 4
−2
4 /5
2
1 /5
−4
1
1 −4
=
=
−4
2
2 −4
=
=
3
1 /10
0
−1
1
8
1
1 /2
1 −1
1
1
2
+ 8
−1
1 /2
1
1 /2
=
=
3
1 /10
0
5
−1
3
1 −3
9
−5
+ 5
−3
9 /10
3
−1
1
−1
1
3
1 /10
3
1 /10
−5
0
−1
1
0 −3
1
1 /2
1 −1
1
1
−5
− 3
1
1 /2
−1
1 /2
1
1
−6
0
−1
1
0 −2
1
1 /2
1 −1
1
1
−6
− 2
−1
1 /2
1
1 /2
1
1
1.3. 2 × 2 symmetric matrices
−4
3
3
4
−4
3
3 −4
=
=
−4
4
4
2
=
=
−4
5
5 −4
=
=
1
6
6
1
=
=
1
6
6 −4
−3
1
1
9 −3
+ 5
3
−3
1 /10
−1
1
−2
1
−1
1
−1
1
−2
3
=
=
29
−5
1
3
−5
0
0
5
−3
1
1
1
−6
0
0
−2
1
4
1
2 /5
4 −2
1
2
−6
+ 4
−2
1 /5
2
4 /5
−9
0
1
1
−5
0
3
2
−8
0
1
2
0
−1
1
1
1
1 /2
1 −1
1
1
−9
+ 1
−1
1 /2
1
1 /2
1
1
0
−1
1
7
1
1 /2
1 −1
1
1
−5
+ 7
−1
1 /2
1
1 /2
0
5
−2
3
1
6
6
6
4 −6
9
−8
+ 5
−6
9 /13
6
=
−3
2
2
3
−3
0
0
10
−3
2
=
2
6
6
2
=
=
2
6
6 −3
9 −6
4
−3
+ 10
−6
4 /13
6
−1
1
1
1
−4
0
−2
3
3
2
−7
0
3
2 /13
6
4 /13
2
3 /13
6
9 /13
0
−1
1
8
1
1 /2
1 −1
1
1
−4
+ 8
1
1 /2
−1
1 /2
=
=
3
9 /10
−7
0
−1
1
0 −1
1
1 /2
1 −1
1
1
−7
− 1
−1
1 /2
1
1 /2
=
=
1
3 /10
0
6
−2
3
4 −6
9
−7
+ 6
−6
9 /13
6
3
2 /13
6
4 /13
30
Functions of matrices
−1
6
6 −1
=
=
−1
6
6
4
0
−1
1
5
1
1 /2
1
1
1 −1
+ 5
−7
1
1 /2
−1
1 /2
−1
1
1
1
−7
0
−3
2
2
3
−5
0
=
0
8
−3
2
=
3
2
2
6
=
=
3
6
6
3
=
=
3
6
6 −2
−2
3
3
6
−2
1
−1
1
2
0
1
1
−3
0
−2
3
3
2
−6
0
4 −6
9
+ 7
−6
9 /13
6
−3
1
1
2
0
−1
1
9
1
1 /2
1 −1
1
1
−3
+ 9
−1
1 /2
1
1 /2
=
−6
=
1
3
−3
0
0
7
0
7
−2
3
−3
1
=
−2
6
6
3
9 −3
1
−3
+ 7
−3
1 /10
3
=
−3
2
2
3
−6
0
0
7
−3
2
=
−2
6
6 −2
=
=
4
6
6 −1
9 −6
4
−6
+ 7
−6
4 /13
6
−1
1
1
1
−8
0
−2
3
3
2
−5
0
3
2 /13
6
4 /13
1
3 /10
3
9 /10
2
3 /13
6
9 /13
0
−1
1
4
1
1 /2
1 −1
1
1
−8
+ 4
1
1 /2
−1
1 /2
=
=
6
9 /13
0
−2
1
7
1
2 /5
4 −2
1
2
2
+ 7
−2
1 /5
2
4 /5
=
9 −6
4
−5
+ 8
−6
4 /13
6
2
3 /13
0
8
−2
3
4 −6
9
−5
+ 8
−6
9 /13
6
3
2 /13
6
4 /13
1.3. 2 × 2 symmetric matrices
4
6
6
4
−3
6
6
2
−1
1
1
1 −1
+ 10
1
−1
1 /2
−3
2
=
=
31
−2
=
1
1
2
3
−2
0
−7
0
0
10
0
6
−1
1
−3
2
=
−3
6
6 −3
=
=
−3
6
6
6
=
=
5
6
6
5
5
6
6 −4
−4
6
6
1
−1
1
−2
1
=
−9
0
1
2
−6
0
−1
1
1
1
−1
0
1 −1
1
+ 11
−1
1 /2
1
−1
2
2
1
−7
0
−3
2
2
3
−8
0
−1
1
1
−4
6
6
5
=
=
−4
6
6 −4
=
=
6
9 /13
0
11
−1
1
1
1 /2
1
1 /2
0
−1
2
8
2
1 /5
1 −2
4
2
−7
+ 8
−2
4 /5
2
1 /5
=
=
2
3 /13
0
−2
1
9
1
2 /5
4 −2
1
2
−6
+ 9
−2
1 /5
2
4 /5
=
=
1
1 /2
0
−1
1
3
1
1 /2
1 −1
1
1
−9
+ 3
−1
1 /2
1
1 /2
=
9 −6
4
−7
+ 6
−6
4 /13
6
1
1 /2
0
5
−3
2
9 −6
4
−8
+ 5
−6
4 /13
6
−2
1
−1
1
−7
0
−10
0
6
9 /13
0
−2
1
8
1
2 /5
4 −2
1
2
−7
+ 8
2
4 /5
−2
1 /5
1
2
2
3 /13
0
−1
1
2
1
1 /2
1 −1
1
1
−10
+ 2
−1
1 /2
1
1 /2
1
1
32
Functions of matrices
6
1
1
6
=
=
6
2
2
3
=
=
6
2
2
6
=
=
6
3
3 −2
−1
1
−1
2
−1
1
−1
3
5
0
0
−1
1
7
1
1 /2
1
1
1 −1
+ 7
5
1
1 /2
−1
1 /2
2
0
1
1
4
0
3
1
−3
0
1
1
0
−1
2
7
2
1 /5
1 −2
4
2
2
+ 7
−2
4 /5
2
1 /5
2
1
0
−1
1
8
1
1 /2
1 −1
1
1
4
+ 8
−1
1 /2
1
1 /2
=
0
7
−1
3
=
6
3
3
6
=
=
6
4
4
6
1 −3
9
−3
+ 7
−3
9 /10
3
−1
1
1
1
3
0
−1
1
1
1
2
0
=
0
10
−1
1
6
5
5
6
1 −1
1
2
+ 10
−1
1 /2
1
=
−1
1
1
1
1
0
0
11
−1
1
=
6
6
6
1
6
6
6 −3
1 −1
1
1
+ 11
−1
1 /2
1
−2
3
4 −6
9
+ 10
6
−6
9 /13
−1
2
=
=
=
=
3
1 /10
0
−1
1
9
1
1 /2
1 −1
1
1
3
+ 9
−1
1 /2
1
1 /2
=
3
1 /10
−3
3
2
0
10
−2
3
1
1 /2
1
1 /2
1
1 /2
3
2 /13
6
4 /13
0
−1
2
9
2
1 /5
1 −2
4
2
−6
+ 9
−2
4 /5
2
1 /5
2
1
−3
0
1
1 /2
−6
0
1.3. 2 × 2 symmetric matrices
1
4
4 −5
=
=
3
3
3 −5
33
−1
2
2
1
−7
0
0
−1
2
3
2
1 /5
4
2
1 −2
+ 3
−7
2
1 /5
−2
4 /5
−1
3
3
1
−6
0
=
=
−2
2
2 −5
=
=
4
6
6 −5
=
=
−5
1
1 −5
=
=
−5
2
2 −2
=
=
−5
2
2 −5
=
=
0
4
−1
3
1 −3
9
−6
+ 4
−3
9 /10
3
−1
2
−1
2
−1
1
−2
1
−1
1
3
1 /10
3
1 /10
−6
0
−1
2
0 −1
2
1 /5
1 −2
4
2
−6
− 1
−2
4 /5
2
1 /5
2
1
0
−1
2
7
2
1 /5
1 −2
4
2
−8
+ 7
−2
4 /5
2
1 /5
2
1
−8
0
−6
0
−1
1
0 −4
1
1 /2
1 −1
1
1
−6
− 4
−1
1 /2
1
1 /2
1
1
−6
0
−2
1
0 −1
1
2 /5
4 −2
1
2
−6
− 1
−2
1 /5
2
4 /5
1
2
−7
0
−1
1
0 −3
1
1 /2
1 −1
1
1
−7
− 3
−1
1 /2
1
1 /2
1
1
34
Functions of matrices
2 × 2 matrices with a repeated eigenvalue
1.4
For each matrix below, the second form can be used to compute functions of matrices, such as
the matrix exponential. For example, if
1 −1
−1
1
2
1
0
1
A =
=
1
3
1
0
0
2
1
1
then
=
A
A
n
2
=
e At
2
n
= e2t
0
1
1
0
0
1
1
0
0
1
1
0
−1 −1
1
1
−1 −1
1
1
−1 −1
1
1
+
+ n2
n−1
t e2t
+
The following 2 × 2 matrices have entries in [−4, 4], and have a repeated integer eigenvalue:
−1 −1
−1
1
0
1
0
1
=
1
1
1
0
0
0
1
1
1
0
−1 −1
=
0
+
0
1
1
1
1 −1
1
3
=
3 −1
1
1
−1 −2
2
3
−2 −2
2
2
2 −1
1
4
1
0
0
1
1
1
1
0
1
0
0
1
−2
2
1
0
1
0
0
1
−2
2
1
0
1
0
0
1
−1
1
1
0
1
0
0
1
2
=
1
=
=
1
0
=
=
−1
1
2
=
=
0
=
=
3
2
0
1
2
+
2
0
+
1
0
0
0
+
3
0
−2 −2
2
2
1
3
−2 −2
2
2
1
0
1 −1
1 −1
1
1
+
+
−1 −1
1
1
1
2
−1 −1
1
1
1
1
0
1
1 −1
0
1
0
2
1
2 /2
0
2
1
2 /2
0
1
1
1
1.4. 2 × 2 matrices with a repeated eigenvalue
−1 −1
4
3
−2 −1
4
2
4 −1
1
2
−2 −3
3
4
−3 −2
2
1
1
0
0
1
−2
4
1
0
1
0
0
1
1
1
1
0
1
0
0
1
−3
3
1
0
1
0
0
1
0
=
3
=
=
1
0
=
=
−2
4
1
=
=
=
1
−2
2
1
0
=
= −1
−3 −3
3
3
=
=
−3 −1
4
1
0
1 −2
2
5
=
1 −1
4
5
3
1
3
1 −1
1 −1
1
0
1
1
0
1
1
2 /4
0
1
1 −1
−3 −3
3
3
0
3
1
3 /3
−1
1
0
0 −1
4
0
−2 −1
+
1
4
2
1
2 /4
0
0
1
0
+
1
0
1
0
0
1
−2
4
1
0
1
0
0
1
=
=
1
0
3
0
−2 −1
4
2
0
4
1
2 /2
−2
2
3
1
2 /4
−1
1
0
0 −1
2
0
−2 −2
+
1
2
2
1
0
+
=
−2 −1
4
2
0
4
1
3 /3
1
0
1
0
0
0
0
3
−3
3
+
1
0
=
1
1
+
−2
4
1
0
+
1
0
= −1
35
3
0
3
0
−2 −2
2
2
1
3
+
−3 −3
3
3
1
3
+
−2 −1
4
2
0
2
1
2 /2
0
4
1
2 /4
36
Functions of matrices
−1 −3
3
5
=
3 −1
1
5
5 −1
1
3
5 −2
2
1
5 −1
4
1
1 −4
1
5
−1 −4
1
3
−2 −4
1
2
−3 −4
1
1
−1
1
1
0
1
0
0
1
1
1
1
0
1
0
0
1
2
2
1
0
1
0
0
1
2
4
1
0
1
0
0
1
−2
1
1
0
1
0
0
1
−2
1
1
0
1
0
0
1
−2
1
1
0
1
0
0
1
3
=
3
=
3
=
1
=
=
0
1
=
=
1
0
4
=
=
=
4
=
1
0
=
=
−3
3
2
=
=
0
−2
1
1
0
=
= −1
2
0
1
2
+
4
0
−3 −3
3
3
1
4
+
4
0
3
0
+
3
0
2 −2
2 −2
1
3
+
1 −1
1 −1
1
3
−1 −1
1
1
1
4
+
2 −1
4 −2
1
3 /3
0
1
1
1
0
1
1 −1
0
1
2 −2 /2
0
1
4 −2 /4
0
1
1
2
0
1
1
2
0
1
1
2
−1
1
0
0 −1
1
0
−2 −4
+
1
1
2
1
2
1
0
3
0
1
3
+
1
0
+
0
0
−2 −4
1
2
1
0
+
−2 −4
1
2
1
1
0
3
−2 −4
1
2
1.4. 2 × 2 matrices with a repeated eigenvalue
−3 −4
4
5
5 −4
1
1
−4 −3
3
2
−4
4
1
0
1
0
0
1
2
1
1
0
1
0
0
1
1
=
=
=
=
3
−3
3
1
0
−4 −4
4
4
=
−4
4
1
0
1
0
0
1
2 −4
1
6
−2
1
1
0
1
0
0
1
−2
2
1
0
1
0
0
1
−2
4
1
0
1
0
0
1
−4
4
1
0
1
0
0
1
−1
1
1
0
1
0
0
1
=
=
2 −2
2
6
2 −1
4
6
=
−2 −4
4
6
=
4 −1
1
6
4
=
=
4
=
4
=
0
2
=
=
5
1
1
3
0
−4 −4
4
4
1
3
+
=
1
0
+
1
0
=
= −1
37
2 −4
1 −2
0
4
0
1
1 −2
−1
1
0
0 −1
3
0
−3 −3
+
1
3
3
0
0
1
0
+
4
0
4
0
+
4
0
+
2
0
5
0
−4 −4
4
4
1
5
−2 −1
4
2
1
2
+
−2 −2
2
2
1
4
−2 −4
1
2
1
4
+
−4 −4
4
4
1
4
+
−1 −1
1
1
1
4 /4
1
3 /3
0
4
1
4 /4
0
1
1
2
0
2
1
2 /2
0
4
1
2 /4
0
4
1
4 /4
0
1
1
1
38
Functions of matrices
6 −1
1
4
=
6 −4
1
2
6 −2
2
2
6 −1
4
2
−4 −5
5
6
−5 −3
3
1
1
0
0
1
2
1
1
0
1
0
0
1
2
2
1
0
1
0
0
1
2
4
1
0
1
0
0
1
−5
5
1
0
1
0
0
1
=
4
=
4
=
=
4
=
1
0
=
=
1
1
5
=
=
1
−5 −4
4
3
−5 −5
5
5
=
1 −3
3
7
=
4
1
4
4
0
2 −2
2 −2
1
4
2 −1
4 −2
0
1
2 −2 /2
0
1
4 −2 /4
−2
1
0
0 −2
3
0
−3 −3
+
1
3
3
1
3 /3
1
0
1
4 /4
1
1
1
0
−5
5
1
0
1
0
0
1
−3
3
1
0
1
0
0
1
0
1
1 −2
1
0
1
0
+
=
2 −4
1 −2
1
5 /5
−4
4
0
4
0
0
1
1 −1
0
5
=
1 −1
1 −1
1
4
+
1
0
= −1
4
0
+
=
+
−3
3
= −2
1
5
+
=
5
0
−5 −5
5
5
−1
1
0
0 −1
4
0
−4 −4
+
1
4
4
0
0
1
0
+
4
0
+
−5 −5
5
5
1
4
−3 −3
3
3
0
5
1
5 /5
0
3
1
3 /3
1.4. 2 × 2 matrices with a repeated eigenvalue
−1 −4
4
7
3 −4
1
7
3 −2
2
7
3 −1
4
7
−3 −5
5
7
5 −1
1
7
7 −1
1
5
7 −4
1
3
7 −2
2
3
−2
1
1
0
1
0
0
1
−2
2
1
0
1
0
0
1
−2
4
1
0
1
0
0
1
−5
5
1
0
1
0
0
1
−1
1
1
0
1
0
0
1
1
1
1
0
1
0
0
1
2
1
1
0
1
0
0
1
2
2
1
0
1
0
0
1
5
=
2
=
6
=
6
=
=
0
1
=
=
1
0
5
=
=
=
5
=
1
0
=
=
−4
4
3
=
=
=
5
=
=
5
39
3
0
1
3
+
5
0
+
5
0
5
0
+
2
0
6
0
+
6
0
+
5
0
5
0
2 −4
1 −2
1
5
1 −1
1 −1
1
5
+
−1 −1
1
1
1
6
−5 −5
5
5
1
6
−2 −1
4
2
1
2
+
−2 −2
2
2
1
5
−2 −4
1
2
1
5
+
+
−4 −4
4
4
1
5
2 −2
2 −2
0
4
1
4 /4
0
1
1
2
0
2
1
2 /2
0
4
1
2 /4
0
5
1
5 /5
0
1
1
1
0
1
1 −1
0
1
1 −2
0
1
2 −2 /2
40
Functions of matrices
7 −3
3
1
=
7 −1
4
3
−5 −6
6
7
−6 −4
4
2
1
0
1
0
0
1
2
4
1
0
1
0
0
1
−6
6
1
0
1
0
0
1
=
5
=
=
3
3
4
=
=
1
−6 −5
5
4
−6 −6
6
6
=
2 −3
3
8
=
−1 −2
8
7
−2 −5
5
8
3
1
5 /5
1
0
1
0
0
1
−3
3
1
0
1
0
0
1
−4
8
1
0
1
0
0
1
−5
5
1
0
1
0
0
1
=
=
−6
6
3
1
0
=
=
1
4 /4
1
0
5
2 −1
4 −2
1
0
1
0
1
1
0
1
4 −2 /4
−2
1
0
0 −2
4
0
−4 −4
+
1
4
4
=
−5
5
0
1
5
+
=
3 −3
3 −3
0
1
3 −3 /3
1
6 /6
1
0
= −1
5
0
0
6
=
+
−4
4
= −2
1
4
+
=
4
0
−6 −6
6
6
−1
1
0
0 −1
5
0
−5 −5
+
1
5
5
0
0
1
0
+
5
0
+
3
0
3
0
−4 −2
8
4
1
3
−3 −3
3
3
1
3
+
+
−6 −6
6
6
1
5
−5 −5
5
5
0
6
1
6 /6
0
3
1
3 /3
0
8
1
4 /8
0
5
1
5 /5
1.4. 2 × 2 matrices with a repeated eigenvalue
−2 −2
8
6
4 −4
1
8
4 −2
2
8
4 −1
4
8
−3 −2
8
5
−4 −6
6
8
−4 −2
8
4
6 −1
1
8
−5 −2
8
3
−2
1
1
0
1
0
0
1
−2
2
1
0
1
0
0
1
−2
4
1
0
1
0
0
1
−4
8
1
0
1
0
0
1
−6
6
1
0
1
0
0
1
−4
8
1
0
1
0
0
1
−1
1
1
0
1
0
0
1
6
=
1
=
2
=
0
=
=
0
1
=
=
1
0
6
=
=
=
6
=
1
0
=
=
−4
8
2
=
=
=
7
−4
8
1
0
=
= −1
41
2
0
1
2
+
6
0
+
6
0
6
0
+
1
0
2
0
+
0
0
+
7
0
−1 −1
1
1
0
8
1
4 /8
0
1
1
2
0
2
1
2 /2
0
4
1
2 /4
0
8
1
4 /8
0
6
1
6 /6
0
8
1
4 /8
0
1
1
1
−1
1
0
0 −1
8
0
−4 −2
+
1
8
4
1
0
−4 −2
8
4
1
7
−6 −6
6
6
1
0
−4 −2
8
4
1
2
−2 −1
4
2
1
1
+
−2 −2
2
2
1
6
−2 −4
1
2
1
6
+
+
−4 −2
8
4
1
6
1
4 /8
42
Functions of matrices
−6 −2
8
2
−4
8
1
0
1
0
−2
1
0
0 −2
8
0
−4 −2
+
1
8
4
1
1
1
0
1
0
0
1
2
1
1
0
1
0
0
1
2
2
1
0
1
0
0
1
3
3
1
0
1
0
0
1
2
4
1
0
1
0
0
1
−7
7
1
0
1
0
0
1
=
= −2
8 −1
1
6
=
=
8 −4
1
4
=
=
8 −2
2
4
8 −3
3
2
=
8 −1
4
4
=
−6 −7
7
8
=
−7 −4
4
1
6
=
=
5
=
6
=
6
=
7
1
−4
4
1
0
−5
5
1
0
=
= −3
−7 −5
5
3
=
= −2
7
0
1
7
+
6
0
1 −1
1 −1
1
6
+
6
0
+
5
0
6
0
+
3 −3
3 −3
1
6
2 −2
2 −2
1
5
+
2 −4
1 −2
1
6
2 −1
4 −2
0
1
1 −1
0
1
1 −2
0
1
2 −2 /2
0
1
3 −3 /3
0
1
4 −2 /4
0
7
1
7 /7
−3
1
0
0 −3
4
0
−4 −4
+
1
4
4
1
4 /4
1
5 /5
1
0
1
0
1
1
+
1
4 /8
−7 −7
7
7
−2
1
0
0 −2
5
0
−5 −5
+
1
5
5
1
0
1.4. 2 × 2 matrices with a repeated eigenvalue
−7 −6
6
5
−6
6
1
0
=
= −1
−7 −7
7
7
=
=
−7 −2
8
1
0
1 −4
4
9
1 −2
8
9
1 −1
9
7
2 −1
9
8
−1 −5
5
9
−1 −1
9
5
0
1
−4
8
1
0
1
0
0
1
−3
9
1
0
1
0
0
1
−3
9
1
0
1
0
0
1
−5
5
1
0
1
0
0
1
−3
9
1
0
1
0
0
1
4
=
=
2
1
4 /8
1
0
=
=
−3
1
0
0 −3
8
0
−4 −2
+
1
8
4
1
0
+
5
1
0
=
=
0
1
1
7 /7
0
0
5
0
+
5
0
4
0
+
5
0
+
4
0
2
0
−5 −5
5
5
1
2
−3 −1
9
3
1
4
+
−3 −1
9
3
1
5
−4 −2
8
4
1
4
−4 −4
4
4
1
5
+
+
−7 −7
7
7
1
5
1
6 /6
0
7
−4
4
4
1
0
=
=
1
0
5
1
0
=
=
−7
7
−1
1
0
0 −1
6
0
−6 −6
+
1
6
6
1
0
5
−4
8
=
=
1
0
=
= −3
43
−3 −1
9
3
0
4
1
4 /4
0
8
1
4 /8
0
9
1
3 /9
0
9
1
3 /9
0
5
1
5 /5
0
9
1
3 /9
44
Functions of matrices
3 −3
3
9
=
3 −1
9
9
−2 −1
9
4
−3 −6
6
9
−3 −1
9
3
−3 −4
9
9
1
0
0
1
−3
9
1
0
1
0
0
1
−3
9
1
0
1
0
0
1
−6
6
1
0
1
0
0
1
−3
9
1
0
1
0
0
1
−6
9
1
0
1
0
0
1
=
1
=
3
=
=
6
=
1
0
=
=
−3
3
6
=
=
0
=
=
3
6
0
1
6
+
6
0
+
1
0
3
0
+
0
0
3
0
−3 −1
9
3
1
3
−6 −6
6
6
1
0
+
−3 −1
9
3
1
3
−3 −1
9
3
1
1
+
+
−3 −3
3
3
1
6
−6 −4
9
6
0
3
1
3 /3
0
9
1
3 /9
0
9
1
3 /9
0
6
1
6 /6
0
9
1
3 /9
0
9
1
6 /9
1.5. 3 × 3 matrices with distinct eigenvalues
1.5
45
3 × 3 matrices with distinct eigenvalues
The following 3 × 3 matrices have nonzero integer entries, are nonsingular, have distinct integer eigenvalues, and are not symmetric:






1
1 −1
4 −2
0
−1
0
0
0 −2
2
 1
1
2  =  −5
1
1  0
1
0   −8 −4
4 
/16
1
2
1
3
1
1
0
0
3
8
10
6






0 −8
8
16
8 −8
0
0
0
4 
10
6 
10 −10 
= −1  0
+ 1  −8 −4
+ 3 8
/16
/16
/16
−8 −4
4
8
10
6
0 −6
6



2
−5



−1
4
=
2
3

5 −5 −5
4
4 
= −1  −4
+
/12
−3
3
3
0
2
0


0
−1


0
4
3
3




2
−3 −3
1
−2

−1  =  5 −2
0  0
2
2
1
1
0


6 −12 −6
15
15 −15



20
10
10
10 −10
= −2 −10
+ 1
/30
−4
8
4
−5 −5
5
0
1
0


0
−2
0   −5
3
9


+ 3

0
2
0


0
−1
0  4
3
3

0
1
0


0
−2
0  5
3
9


+ 3
1
1
1

1
1
1
1
1
−1



2
4
4
0
8
8
0
−4
−4 
+ 3
/12
0
1
−1
−1



1
−5

−1  =  3
1
4

5 −5 −5
3
3 
+
= −1  −3
/12
−4
4
4
1
1
1

1
−1


0
0
1
0
1
1
0
2
2
−1


1
−1
1  0
0
0
1
0
1

2
4
0
4
−4
0
−4
8
0 
+ 3
/12
8



1
−3
3
1
−2

−1  =  2 −1
1  0
−1
5
2
0
0


15 −15
15
6 −6 −12
5 −5
4
8 
= −2  −4
+ 1  −5
/30
10 −10
10
−10
10
20
1
1
1


1
1
1
2
2
−1

2
−1
2

= −3 
/30
/30

1
−4 
/12
9

−3
9
0
0 
/12
−3
9
1
8
−3
3
0
3

2
5 
/30
21

−3
21
0
0 
/30
−3
21
4
−5
−3
9
0
9
1
−4
9
3
3
0

1
8 
/12
−3

9 −3
9 −3 
/12
0
0

4
5 
/30
−3

21 −3
21 −3 
/30
0
0
2
−5
21
9
9
0




2
0 −2
2
−3
0
0
0
2  =  −1
1
1   0 −1
0   −2
−1
1
1
1
0
0
3
2




0
0
0
4 −4 −4
4
0
4 −4 
2
2 
− 1  −2
+ 3 2
/8
/8
0 −4
4
−2
2
2
2
−4
2
2
4
2
2

4
2 
/8
2

4
2 
/8
2
46


Functions of matrices
1
1
1
−1
1
2

= −1 



1
−4 −2
0
−1
2  =  −3
1
1  0
1
5
1
1
0


0
8 −8
16 −8
8
0
6 −6 
4 −4
+ 1  −8
/16
0 −10
10
−8
4 −4
0
1
0


0
0
0   −8
3
8


+ 3

1
1
2



2
−5
1
1
−1

−1  =  4 −2
0  0
1
3
1
1
0



10
0 −10
−4 −8
4
0
8 
16 −8
= −1  −8
+ 1 8
/16
−6
0
6
−4 −8
4
0
1
0


1
1
2
0
2
0

1
1
1



1
−1 −1
1
−1

−1  =  1 −2 −1   0
2
1
1
3
0


5 −4 −3
4
4
4
3 
8
= −1  −5
+ 2 8
/12
−5
4
3
−4 −4
1
1
−1


1
1
2

1
1
2
1
2
1
1
1
2
1
2
−1

/16
−2
4
6
0
8
8

2
−4 
/16
10

0
0
6
10 
/16
6
10

0
−2
0
0   −4 −8
3
10
8

10
8

0
+ 3 0
/16
10
8

2
4 
/16
6

6
0 
/16
6

0
−5
0   −4
3
3

0
0
0

3
0 
/12
3

3
−3 
/12
9
4
−4
0
0
3
0 
+ 3  −3
/12
0
9



2
−1 −1
5
−1
1  =  0 −2
4  0
1
1
1
3
0


−8 −6
−4
8 −4
0
0 
16 −8
+ 1  −8
/16
8
6
4 −8
4
0
1
0


0
−10
8
0   4 −8
3
2
0

10
0

0
+ 3 8
/16
6
0

6
4 
/16
2

10
8 
/16
6



2
−7
1
1
−1

−1  =  4 −1
0  0
1
5
1
1
0



7
0 −7
−4 −12
4
0
4 
12 −4
= −1  −4
+ 2 4
/12
−5
0
5
−4 −12
4
0
2
0


0
−1
0
0   −4 −12
3
9
12

9
12

0
+ 3 0
/12
9
12

1
4 
/12
3

3
0 
/12
3

0
2
0


0
−7
0  4
3
3


5
4 
/12
3

9
12 
/12
3
1

1
−1

10
= −1  0
−10

= −1 
7
0
−7



2
−1 −1
3
−1
1  =  0 −3
4  0
1
1
1
1
0


−4 −5
−4
4 −4
0
0 
12 −12
+ 2  −12
/12
4
5
4 −4
4

9
+ 3  12
/12
3
4
−4
0
0
0
0
1.5. 3 × 3 matrices with distinct eigenvalues
47

1
1
2



1
−1 −1
3
−2

−1  =  2 −1 −1   0
2
1
1
7
0


6 −10 −4
15
10 −5
20
8 
10 −5
= −2  −12
+ 1  15
/30
−6
10
4
−15 −10
5
0
1
0


1
1
2



2
−3
1
1
−1

−1  =  2 −1
0  0
1
1
1
1
0



3
0 −3
−2 −4
2
0
2 
4 −2
= −1  −2
+ 1 2
/4
−1
0
1
−2 −4
2
0
1
0


1
1
2



2
−8
0
1
−2

−1  =  5 −1
0  0
1
7
1
1
0


8 −8 −8
0
0
0
5
5 
15 −5
= −2  −5
+ 2 5
/20
−7
7
7
−5 −15
5
0
2
0


1
1
2



2
−2
2
1
−1

−1  =  1 −1
0  0
1
1
2
1
0



2
0 −2
−2 −6
2
0
1 
3 −1
= −1  −1
+ 2 1
/3
−1
0
1
−2 −6
2
0
2
0


0
1
0


5
5 
/15
0

0
0 
/15
0

0
0 −4
0   −15 −10
3
15
12

0
0

12
+ 3  15
/30
30
24

2
5 
/30
9

0
9 
/30
18
1
−1
−1

2
1
2
2
1
−1

2
2
2


2
−2
2

−1  =  1 −1
−1
2
1



6
2 −10
5 
= −2  −3 −1
+ 1
/15
−6 −2
10


1
1
2
1
1
2
2
2
2
2
−1
2

= −2 

3
−2
1  0
2
0
0 −20
0
10
0 −10

0
−6
10
0   −15 −10
3
3
0

9
0

0
+ 3  −3
/30
21
0

0
−1
0   −2
3
3

3

+ 3 0
/4
3
0
−4
4
4
0
4

1
2 
/4
1

1
0 
/4
1

0
−1
1
0   −5 −15
3
12
8

12
8

0
+ 3 0
/20
12
8

1
5 
/20
8

8
0 
/20
8

0
−1
0   −1
3
3

3

+ 3 0
/3
3

1
1 
/3
0

0
0 
/3
0
0
−3
6
6
0
6

0
−3 −1
0   0 −10
3
3
6

10
9
18
−5 
6
+ 3 3
/15
5
6
12



−1
5 −2
0
−2
2  =  −7
1
1  0
2
1
2
2
0


0 −20
10
30
20 −10
0
28 −14 
5
+ 1  −15 −10
/30
0 −4
2
−30 −20
10

4
5 
/30
3

9
−3 
/30
21
0
1
0

48


Functions of matrices
1
1
2
−1
1
1

= −1 

6
0
−6



2
−1 −1
3
−1
1  =  0
2
4  0
1
1
1
5
0


8 −10
4 −8
4
0
0 
16 −8
+ 1  −8
/16
−8
10
−4
8 −4


2
−3

−1  =  4
1
5

6
0 −6
0
8 
= −1  −8
+
/16
−10
0
10


1
1
2
1
1
2
−1
1
−1

−1
2
1

= −1 

5
0
−5

1
−1
0  0
1
0
1
2
1

1
4
8
4
8
16
8
1
1
1
2
4
4
4

1
1
0

0
−1
1  0
1
0
−1
2
−1



−1
4
2  =  −5
2
3


4 −4
4
5 −5 
= −1  −5
+
/12
3 −3
3
1
 1
−1

1
1
1

1
−1
0  0
1
0


2
8
8
0
12
12
12
4
4
0
1
1
2

+ 3
/16
6
8
10
0
1
0


0
−2
0  4
3
6

0
2
0


0
−5
0   −4
3
3


−8
8
0

10
−4 
/16
2

0
6
0
8 
/16
0
10

2
−4 
/16
10

−8
10
0
0 
/16
−8
10
0
8
−8
6
0
6
−4
4
0
3
+ 3  12
/12
9

7
−4 
/12
3

0
3
0
12 
/12
0
9


0
−1
0
1
0  4
12 −4 
/12
3
3 −12
9


−4
3 −12
9
−4 
0
0 
+ 3 0
/12
/12
−4
3 −12
9
0
2
0

0
2
0


0
1
0  8
3
−3

−4
0
−4 
+ 3  −3
/12
0
−3



1
0 −2
4
−1
2  =  −1 −1
5  0
1
1
1
3
0



0
0
0
16 −8 −8
10 −6 
= −1  −8
+ 1  8 −4 −4
/16
8 −10
6
−8
4
4
1
 1
−1

0
−6
0   −4
3
2

−4
−8 
+ 3
/16
−4



2
−1 −1
1
−1
1  =  0
3
4  0
1
1
1
3
0


4 −7
4 −4
4
0
0 
12 −12
+ 2  −12
/12
−4
7
−4
4 −4


2
−5

−1  =  4
1
7

5
0 −5
0
4 
= −1  −4
+
/12
−7
0
7
1
1
2
0
1
0

−1
4
3

1
−4 
/12
9

0
0
3
9 
/12
3
9

0
8 −10
0   −8
4
3
0
2

0
8

10
+ 3 0
/16
0
6
0
1
0


6
4 
/16
2

8
10 
/16
6
1.5. 3 × 3 matrices with distinct eigenvalues

49



−1
1 −2 −1
−1
1  =  −1 −1
1  0
2
1
1
3
0



4 −5
3
8
8
5 −3 
4
= −1  −4
+ 2 4
/12
4 −5
3
−4 −4
0
2
0


0
4
0   −4
3
0

0
2
0


0
4
0   −4
3
0




−1
4 −1
0
−1
1  =  −3
0
1  0
1
5
1
1
0



4 −4
4
8
4 −4
3 −3 
0
0
= −1  −3
+ 2 0
/12
5 −5
5
−8 −4
4
0
2
0


0
1
0   −8
3
3


+ 3

0
1
0

−1
3
2

0
2
0  0
2
−2

4

+ 2 2
/6
−2
1
3
−4

0
2
0  0
2
−2

9
4
3 
+ 2 2
/6
3
−2
−5
3
2
1
 1
−1
1
1
2




−1
1 −2
1
2  =  −1 −1
3 
1
1
1
1



4 −3
5
8
3 −5 
= −1  −4
+ 2 4
/12
4 −3
5
−4
1
 1
−1
1
2
1

1
 1
−1
−1
0
0
0
0
0
−8
−4 
+ 3
/12
4
1
2
2
1
 1
−1
1
2
2

= −1 

0
0 
+ 3
/12
0


−1
2
−1  =  1
−1
5

2 −2
2
1 −1
1 
+
/6
5 −5
5

1
−1
2  0
1
0
0
1
1

0
1  −9
−9
0
3
3

1
2
−1



−1
1
−1  =  −1
2
1

2 −5 −1
5
1 
= −1  −2
+
/6
2 −5 −1
1
 1
−1
1
−1
2


−2
−1
−1   0
1
0
3
1
1

1
0
0
0
9
3
3
0
0
0
0
0
0
0
3
3

3
0 
/12
3

−3
3 
/12
9
3
9
3

5
4 
/12
3

3
9 
/12
3
0
9
9

1
4 
/12
3

0
3 
/12
3
−1
−4
9
0
1
0

0
1
0

−3
3
9
−3
0
3

0
1
0   −9
2
4

0
4
3 
+ 2 8
/6
3
4



−1
1 −3 −2
−1
2  =  −1
1 −1   0
−1
1
1
1
0


2
1
5
0 −9 −9
3
3
= −1  −2 −1 −5 
+ 1 0
/6
2
1
5
0
3
3
1
 1
−1
/12
−5
−4
3
2
4
2
8
4
−4
−4
−2
2

1
3 
/6
−2

−2
−4 
/6
−2

5
3 
/6
−2

4
2 
/6
−2

−1
3 
/6
4

−8
−4 
/6
4
50
Functions of matrices



−1
−2
2  =  5
2
1

2 −2
2
5 −5 
= −1  −5
+
/6
−1
1 −1
1
 1
−1
1
−1
−1


−1
−1
1  0
2
0

0
−1
0  9
2
−4

0
4
3 
+ 2  −4
/6
3
−8
0
1
0

1
3
−2

0
4
0   −4
3
0

0

+ 3 0
/8
0
−6
4
2
0
2
0


0
5
0   −5
3
0

−8
0
4



−1
5 −1
1
−1
1  =  −3
0
1  0
1
4
1
0
0



5 −5
5
4 −4 −8
3 −3 
0
0
= −1  −3
+ 2 0
/12
4 −4
4
−4
4
8
0
2
0


0
1
0   −4
3
3


+ 3




−1
3 −3
1
−2
1  =  −2
1
1  0
−1
5
2
0
0



6 −6
12
15 −15 −15
4 −8 
5
5
= −2  −4
+ 1  −5
/30
10 −10
20
−10
10
10
0
1
0


0
2
0   −5
3
9


+ 3

0
1
0


0
8
0   −8
3
0


+ 3

1
 1
−1
2
1
2

0
= −1  −4
4

0
1
1

1
0
9
9



2
0 −2
4
−1
2  =  −1 −1
3  0
1
1
1
1
0


0
0
8 −8 −8
6 −2 
+ 1  4 −4 −4
/8
−6
2
−4
4
4



−1
1 −3
2
2  =  −1 −1
3 
1
1
1
2



5 −8
7
15
8 −7 
= −2  −5
+ 2 5
/20
5 −8
7
−5
1
 1
−1

1
 1
−1
1
 1
−1
2
1
2
0
3
3
−2
0
0
0 −15
0 −5 
+ 3
/20
0
5
2
2
1
2
2
1



−1
0 −2 −4
−1
2  =  −1 −1
3  0
1
1
1
5
0


0
0
0
16
8
8
6 −10 
4
4
= −1  −8
+ 1 8
/16
8 −6
10
−8 −4 −4
1
 1
−1
−1
1
2

0
1
0

/12
/30
/16
0
0
0
2
−2
−4
8
6
2
8
12
8

7
5 
/20
4

8
12 
/20
8
9
9
0
21
21
0

4
5 
/30
3

3
3 
/30
0
3
3
0
−2
5
21

10
−4 
/16
2

−8 −8
6
6 
/16
10
10
−6
−4
2
0
0
0

2
4 
/8
2

8
6 
/8
2

1
8 
/12
3

3
3 
/12
0
−1
4
9
9
9
0

−1
3 
/6
2

−2
2 
/6
4
1.5. 3 × 3 matrices with distinct eigenvalues




1
−1
1 −1

−1  =  1 −1 −3  
1
1
2
1


5 −3 −4
4
3
4 
= −1  −5
+ 2  −4
/12
−5
3
4
8
51
0
2
0


0
−5
0  4
3
−3

3
0
−3



1
−1
1 −3
−2

−1  =  1 −1 −7   0
−1
2
1
1
0


6 −4 −10
15 −5
10
4
10 
5 −10
= −2  −6
+ 1  −15
/30
−12
8
20
15 −5
10
0
1
0


0
−6
0   15
3
−3

4
−5
−3

1
2
1



1
−1
1
5
−1

−1  =  1 −1
3  0
1
0
2
4
0



10 −6 −8
−4 −4
8
6
8 
4 −8
= −1  −10
+ 1 4
/16
0
0
0
−8 −8
16
0
1
0


1
2
1



1
−1
1
3
−1

−1  =  1 −1
1  0
2
0
3
4
0



7 −5 −4
−4 −4
4
5
4 
4 −4
= −1  −7
+ 2 4
/12
0
0
0
−12 −12
12




−1
−1 −1
3
−1

1  =  1
1
5  0
1
0
2
4
0



6 −10
8
4
4 −8
10 −8 
8
= −1  −6
+ 1  −4 −4
/16
0
0
0
−8 −8
16
0
1
0



1
2
1
1
2
1
1
2
1
1
2
−1

−1
0
0
0
0
0
4
3
−4 
+ 3 9
/12
8
−3
1
2
−1

2
1
1
2
1
1
2
1
1



−1
−1 −1
1
−1

1  =  1
1
3  0
2
0
3
4
0



5 −7
4
4
4 −4
7 −4 
4
= −1  −5
+ 2  −4 −4
/12
0
0
0
−12 −12
12
1
2
1
2
1
1

9
+ 3  21
/30
−3
3
9
−3

4
4 
/12
0

0
0 
/12
0

10
10 
/30
0

9
0
21
0 
/30
−3
0

0
−10
6
0   −4 −4
3
2
2

10
10

6
+ 3 6
/16
8
8

8
8 
/16
0

0
0 
/16
0
0
2
0


0
−7
0   −4
3
3


+ 3

4
4 
/12
0

0
0 
/12
0
/12
9
3
12
5
−4
3
9
3
12

0
−6
0   −4
3
2

10
−4
2
0
2
0


0
−5
0   −4
3
3

7
−4
3

+ 3

6
+ 3  10
/16
8
/12
3
9
12

−8
8 
/16
0

6
0
10
0 
/16
8
0

−4
4 
/12
0

3
0
9
0 
/12
12
0
52
Functions of matrices

1
2
1



2
1
−5 −1
1
−1

1
1  =  3 −1
1  0
−1
1
4
2
0
0



10 −10
0
−4
4 −8
6
0 
4 −8
= −1  −6
+ 1  −4
/16
−8
8
0
8 −8
16
0
1
0


1
2
1



1
−7 −1
1
−1
1  =  5 −1
1  0
2
4
1
0
0


−7
0
−4
4 −12
5
0 
4 −12
+ 2  −4
/12
4
0
4 −4
12
0
2
0


0
−1
0  4
3
9


+ 3
2
1
−1




2
−3 −1
1
−1
2  =  1 −1
1  0
1
2
1
0
0


−3
0
−2
2 −4
1
0 
2 −4
+ 1  −2
/4
2
0
2 −2
4
0
1
0


0
−1
0  2
3
3

3

+ 3 3
/4
0
1
−2
1
2
1
−1




2
−2 −2
1
−1
2  =  1 −2
1  0
2
1
1
0
0


−2
0
−2
2 −6
1
0 
2 −6
+ 2  −2
/3
1
0
1 −1
3
0
2
0


0
−1
0  1
3
3

3

+ 3 3
/3
0
1
−1
0

2
1
−1

7
= −1  −5
−4


1
2
1
3
= −1  −1
−2


1
2
1
2
= −1  −1
−1

0
−2
2
0   4 −4
3
10
6

10
6

6
+ 3  10
/16
0
0

1
2
1



2
−8
0
1
−2

−1  =  7 −1
1  0
1
5
1
0
0


8 −8 −8
0
0
0
7
7 
5 −15
= −2  −7
+ 2  −5
/20
−5
5
5
5 −5
15
0
2
0


1
2
1
0
1
0

2
1
−1



−1
−3

−1  =  5
1
4

6 −6
0
10
0 
= −1  −10
+
/16
−8
8
0
2
1
−1


1
−1
1  0
0
0
1
1
2

1
4
4
8
−4
−4
−8
/12
1
−4
3
9
9
0

0
8 
/16
8

8
8 
/16
0

0
12 
/12
12

3
12
3
12 
/12
0
0
1
1
0
0
0
0

0
4 
/4
4

4
4 
/4
0

0
3 
/3
6

6
6 
/3
0


0
−1
1
1
0   5 −5
15 
/20
3
12
8
8


12
8
8

8
8 
+ 3  12
/20
/20
0
0
0

0
−2
0  4
3
6

8
8 
+ 3
/16
16

0
8 
/16
−8

10 −8
10 −8 
/16
0
0
2
−4
10
6
6
0
1.5. 3 × 3 matrices with distinct eigenvalues



−1
−5

−1  =  7
2
4

5 −5
0
7
0 
= −1  −7
+
/12
−4
4
0
1
2
1

2
1
−1


1
−1
1  0
0
0
1
1
1

2


2
−2 −2

2  =  2 −1
2
1
1


6 −10
2
10 −2 
= −2  −6
+ 1
/15
−3
5 −1
1
2
1

1
2
1

2
−1
−1

−1
2
2

= −2 


1
2
2


10 −20
5 −10 
+ 3
/15
−5
10
5
−5
0
9
6
3

−1
10 
/15
6

0
18
0
12 
/15
0
6

0
0
0   −4
3
4

0

+ 3 4
/4
4
−1
−2
3



2
−1 −1
3
−1
2  =  0 −2
4  0
1
1
1
1
0


−4 −2
−4
4 −4
0
0 
8 −8
+ 1  −8
/8
4
2
4 −4
4
0
1
0


0
−6
0  4
3
2

6

+ 3 8
/8
2
4
−4
0
1
1
−1

6
0
−6
1
1
−1


0
−3
0  0
3
3

0
1
0



2
−1 −1

−1  =  1 −3
1
1
1


8 −5 −7
5
7 
= −2  −8
+ 2
/20
−8
5
7
1
2
2

3
−2
2  0
1
0
0
0
0
0
1
0

12
12 
+ 3
/12
12

1
0
−4
12 
/12
9 −12

3
9 −12
3
9 −12 
/12
0
0
0



−1
2 −1
0
−1
2  =  −3
1
1  0
1
1
1
1
0


0 −2
2
4
2 −2
0
3 −3 
2
+ 1  −4 −2
/4
0 −1
1
−4 −2
2
1
1
2
= −1 
−4
−4
−4

0
−1
0  4
3
3

0
1
0

= −1 
1
2
2
4
4
4
0
2
0




2
−5 −2
0
−2
2  =  −1
2
2  0
−1
7
1
1
0


0
10 −20
30 −10
20
0
2 −4 
10 −20
+ 1  −30
/30
0 −14
28
−15
5 −10


53

3
−2
2  0
2
0
0
0
0

0
0
0   −15
3
15


−2
4
5 −10 
/30
9
12

0
0
0

18
24 
+ 3  30
/30
/30
15
9
12
0
3
3
0
0
0

0
−8
5
0   0 −5
3
4
0

5 −5
12
0
15 −15 
0
+ 3 8
/20
−5
5
8
0
0
2
0


1
2 
/4
1

0
1 
/4
1

2
4 
/8
2

6
8 
/8
2

7
5 
/20
4

12
8 
/20
8
54
Functions of matrices




2
−1 −1
1
−1
2  =  0 −3
2  0
1
1
1
0
0


−1 −1
−2
1 −2
0
0 
3 −6
+ 2  −6
/3
1
1
2 −1
2
0
2
0




2
−3
0
1
−2

2  =  −1 −2
2  0
−1
5
1
0
0


6 −3 −6
0
0
0
10 −10
= −2  2 −1 −2 
+ 1  −20
/15
−10
5
10
10 −5
5
0
1
0


1
2
2
1
2
−1

= −1 

1
2
2


1
2
2
1
−1
0

0
−2
0   10
3
9

1
−5
3




1
0 −1
1
−3
0
0
0
2  =  −1
1
1   0 −1
0   −2
−1
1
1
1
0
0
3
2




0
0
0
2 −1 −1
2
0
2 −2 
1
1 
− 1  −2
+ 3 2
/4
/4
0 −2
2
−2
1
1
2
−2
1
1
2
0
−2
1
2
−1

1
−1
2

= −3 


0
−2
0  2
3
3

3

+ 3 6
/3
0

0
0
0
9
+ 3  18
/15
0
3
6
0
1
1
1



1
−1
1
3
−1

−1  =  1 −1
1  0
1
0
2
4
0



6 −2 −4
−4 −4
4
2
4 
4 −4
= −1  −6
+ 1 4
/8
0
0
0
−8 −8
8
0
1
0


0
−6
0   −4
3
2

6

+ 3 2
/8
8
2
−4
2




1
−1
1
1
−1

−1  =  1 −1
0  0
2
0
3
2
0



2 −1 −1
−2 −2
1
1
1 
2 −1
= −1  −2
+ 2 2
/3
0
0
0
−6 −6
3
0
2
0


0
−2
0   −2
3
3

3

+ 3 0
/3
6
1
−2
3

0
1
0


0
−12
0   −5
4
2

3
−5
2

+ 4
1
2
2
1
2
2
2
1
2
2
1
2



2
−1
1
4
−1

−1  =  1 −2
1  0
2
0
2
5
0



12 −3 −9
−5 −5
5
3
9 
10 −10
= −1  −12
+ 1  10
/15
0
0
0
−10 −10
10
1
2
2
2
1
2
/15
8
2
10
6
2
8
3
0
6
8
2
10

1
2 
/3
3

3
6 
/3
0

2
5 
/15
6

6
12 
/15
0

2
1 
/4
1

1
1 
/4
1

4
4 
/8
0

0
0 
/8
0

1
1 
/3
0

0
0 
/3
0

9
5 
/15
1

4
1 
/15
5
1.5. 3 × 3 matrices with distinct eigenvalues

55



−1
−1 −1
1
−1

1  =  1
1
3  0
1
0
2
4
0



2 −6
4
4
4 −4
6 −4 
4
= −1  −2
+ 1  −4 −4
/8
0
0
0
−8 −8
8
0
1
0


0
−2
0   −4
3
2

2

+ 3 6
/8
8
6
−4
2




−1
−1 −1
0
−1

1  =  1
1
1  0
2
0
3
2
0



1 −2
1
2
2 −1
2 −1 
1
= −1  −1
+ 2  −2 −2
/3
0
0
0
−6 −6
3
0
2
0


0
−1
0   −2
3
3

0

+ 3 3
/3
6
2
−2
3




−1
−1 −2
1
−1

2  =  1
1
4  0
2
0
2
5
0



3 −12
9
10
10 −10
12 −9 
5
= −1  −3
+ 1  −5 −5
/15
0
0
0
−10 −10
10
0
1
0


0
−3
0   −5
4
2

12
−5
2

+ 4

0
2
0

1
2
2
1
2
2
1
2
2
2
1
2
2
1
2
2
1
2


1
−1
1

−1  =  1 −1
1
1
3


8 −7 −5
7
5 
= −2  −8
+ 2
/20
−8
7
5
1
2
2
2
1
−1


3
−2
2  0
2
0
0 −5
0
5
0 −15
0
3
6
1
2
2



2
2
−4 −1
2
−1

1
2  =  1 −1
2  0
−1
2
3
1
1
0



12 −12
0
−5
10 −10
3
0 
10 −10
= −1  −3
+ 1  −5
/15
−9
9
0
5 −10
10
0
1
0


1
2
2
0
1
0




2
−4
1
2
−1

−1  =  3 −1
1  0
1
1
1
2
0



12
0 −12
−5 −10
10
0
9 
10 −10
= −1  −9
+ 1 5
/15
−3
0
3
−5 −10
10

−1
1 
/3
0

0
0 
/3
0

−9
5 
/15
1

2
1
8
4 
/15
10
5

0
−8
7
0   0 −5
3
4
4

5
12
12
−5 
8
+ 3 8
/20
15
8
8

2
2
2
/15
2
8
10
2
6
8

−4
4 
/8
0

0
0 
/8
0

5
5 
/20
0

0
0 
/20
0


0
−3
3
0
0   5 −10
10 
/15
4
4
1
5


8
2
10

2
10 
+ 4 8
/15
/15
4
1
5


0
−3
0
3
0   −5 −10
10 
/15
4
4
5
1


8
10
2

5
1 
+ 4 4
/15
/15
8
10
2
56
Functions of matrices




2
−1 −1
4
−1
2  =  0 −2
5  0
1
1
2
1
0


−9 −3
−5
5 −5
0
0 
10 −10
+ 1  −10
/15
9
3
10 −10
10
1
2
2
2

2
−1

12
= −1  0
−12

1
2
2

2
−1
1

= −2 

0
1
0



0
−12
0  5
4
2

9
−5
1

0
0
0   −4
2
8

3
−4
5
8
+ 4  10
/15
2



−1
−3 −1
1
−2
0
1  =  5
2
1   0 −1
−1
1
2
1
0
0



0 −9
9
4
4 −8
0
15 −15 
16 
− 1  −8 −8
/12
/12
0
3 −3
−8 −8
16
+ 2
4
5
1
8
8
8

−3
8 
/12
−1

5 −1
5 −1 
/12
5 −1



1
3
0
1
−2

−1  =  −5 −1
0  0
2
1
2
2
0


6 −6 −3
0
0
0
10
5 
5 −5
= −2  −10
+ 1  10
/15
2 −2 −1
−20 −10
10
0
1
0


0
2
0   −10
3
9


+ 3




2
−1 −1
1
−1
2  =  0
2
4  0
1
1
1
3
0


4 −6
4 −4
4
0
0 
8 −8
+ 1  −8
/8
−4
6
−4
4 −4
0
1
0


0
−2
0   −4
3
2

2

+ 3 8
/8
6
−4
4
0



1
−2 −1
0
−1
2  =  −1
1
1  0
1
3
1
1
0


2 −2
4 −2
2
1 −1 
2 −2
+ 1  −4
/4
−3
3
−4
2 −2
0
1
0


0
0
0   −4
3
4

0

+ 3 4
/4
4
−1
2
1
1
2
2

1
2
2
2
−1
2

−1
1
1

= −1 


1
2
2
2
0
−2
−1
1
2

= −1 
0
0
0
/15
9
0
18

3
5 
/15
2

8
10 
/15
2

−1
5 
/15
3

6
3
0
0 
/15
12
6
−2
−5
6
0
0
0
0
1
1

6
−4 
/8
2

2
8 
/8
6

1
−2 
/4
3

0
3 
/4
3
1.6. 3 × 3 singular matrices
1.6
57
3 × 3 singular matrices
The following 3 × 3 matrices have nonzero integer entries, are singular, have distinct integer
eigenvalues, and are not symmetric:






1
1
2
−2 −3
1
−1
0
0
0 −4
4
 1
1
2  =  −2
1
1  0
0
0   −5
5
0 
/20
1
2
1
3
1
1
0
0
4
5
7
8






0
8 −8
15 −15
0
5
7
8
5
0 
7
8 
8 −8 
= −1  0
+ 0  −5
+ 4 5
/20
/20
/20
−5
5
0
5
7
8
0 −12
12


1
1
1


2
−3
1


1
1 −2
=
2
1
1


0 −9
0
3 
+ 1
/12
0
3
1
2
1

9
= 0  −3
−3


1
1
1
2
1
2

0
0
0
= −1 


1
1
1
2
2
1

9
= 0  −3
−3


1
1
2
1
1
1

= −1 


1
1
2
8
8
−12
1
1
2

= −1 
0
8
−8

1
1 
1
0
0
0
−4
8
−4



1
2 −3
1
2  =  −3
1
1 
1
2
1
1


−8
8
15


12 −12
+ 0 −5
/20
−8
8
−5


1
−3 −1
1  =  1 −1
2
1
2


−9
0
3
0 
+ 1
/12
3
0

1
1 
1
0
0
0
4
4
−8

0
−3


0
0
4
3


4
3
−8 
+ 4 3
/12
4
3
0
−4
4

0
0
0   −5
4
5

−4
0
8
0
0
0
0
1
0
−1
0
0
0
0
0

0 −15
0
5 
+ 4
/20
0
5
4
4
4
5
5
5

0
−3
0  0
4
3


−4
3
−4 
+ 4 3
/12
8
3
0
0
0



2
−2
1
1
−1
2  =  −2 −3
1  0
1
3
1
1
0


5 −5
0 −8
15
0 −8 
+ 0  −15
/20
5 −5
0
12
0
1
0
0
0
0


0
−4
0  5
4
7

0
0
0


0
−8
0   −15
4
1


+ 4
0
0 
+ 4
/20
0



1
0 −1
5
−1
2  =  −1
1
7  0
1
1
0
8
0


0
0
15 −5 −5
8 −12 
5
5
+ 0  −15
/20
−8
12
0
0
0
/20

3
4 
/12
5

5
5 
/12
5
8
8
8

0
4 
/12
4

4
4 
/12
4
3
−4
5
5
5
5
0
−5
5
7
7
7
5
5
5
−8
5
1
5
7
8

4
5 
/20
7

7
7 
/20
7

4
0 
/20
8

8
8 
/20
8

12
5 
/20
1

5
5
7
7 
/20
8
8
58


Functions of matrices
1
1
2
1
1
2

= −1 


1
1
2
1
2
1

= 0


1
1
2
= −1 

9
0
−9
2
1
1


8
0
−8
1
1
2
12
0
−12
2
2
2

= −1 
5
5
−10



2
−1 −1
7
−1
1  =  0
1
5  0
1
1
0
8
0


8 −12
5 −15
5
0
0 
15 −5
+ 0  −5
/20
−8
12
0
0
0



1
−1
0
3
1  =  0 −1
4 
2
1
1
5


−3 −3
0
0
0
0 
8
+ 1  −4
/12
3
3
4 −8
0
0
0


0
−8
0   −5
4
1


+ 4
0
1
0
0
0
0




2
−1
2
1
−1
2  =  −1 −3
1  0
1
2
2
1
0


0 −5
6 −6
0 −5 
9
+ 0  −9
/15
0
10
6 −6
0
0
0

6
6
6

5
0 
/15
5

5
5 
/15
5

0
8 −12
0   −15
5
4
1
1

5
5

8
+ 4 8
/20
7
7

8
5 
/20
1

5
8 
/20
7
0
0 
+ 5
/15
0



1
0 −1
5
−1

2  =  −1
0
8  0
1
1
1
7
0



0
0
0
15 −5 −5
12 −8 
0
0
= −1  −8
+ 0 0
/20
8 −12
8
−15
5
5
0
0
0


1
2
1
1
2
1
0
1
0
9
= 0  −9
0
3
4
5

0
−5
0  3
5
4

1
1
2


3
4 
/12
1

3
4 
/12
5
3
−8
1

8
5 
/20
1

8
5 
/20
7
1
2
1

0
−9
0   −4
4
1


0
3
−8 
+ 4 5
/12
8
4
0
0
0

12
−5 
/20
1

7
7
5
5 
/20
8
8

0
−12
8
0   5 −15
4
1
1

8
8

5
+ 4 5
/20
7
7


7
5
8

0
−9
0  4
4
1


0
3
−4 
+ 4 4
/12
4
5
0
0
0



2
−1
0
8
−1
1  =  0 −1
5  0
1
1
1
7
0


−8 −8
0
0
0
0
0 
15 −5
+ 0  −5
/20
8
8
5 −15
5



1
−1
0
3
1  =  1 −1
5 
2
0
1
4


−3 −3
0
0
3
3 
4
+ 1 4
/12
0
0
−4 −4
/20
−8
15
1
0
−3
6
4
4
4
3
−4
1
3
5
4

3
8 
/12
1

3
5 
/12
4
1.6. 3 × 3 singular matrices

59



1
−1 −1
7
−1

2  =  1
0
8  0
1
0
1
5
0



8 −12
8
5
5 −15
12 −8 
0
0
= −1  −8
+ 0 0
/20
0
0
0
−5 −5
15
0
0
0


0
−8
0   −5
4
1


+ 4




2
−1
0
8
−1
1  =  1 −1
7  0
1
0
1
5
0


−8 −8
0
0
0
8
8 
5 −15
+ 0 5
/20
0
0
−5 −5
15
0
0
0


0
−12
0   −5
4
1


+ 4



1
2 −1
1
1  =  −3 −1
1 
1
2
3
1


−8
0
5
12
0 
+ 0 5
/20
−8
0
−15
0
0
0


0
4
0   −5
4
7

0
0
0


0
5
0   −3
5
4

0
0
0


0
−4
0  0
4
8

0
0
0


0
−5
0   −6
5
1


+ 5

1
2
1
2
1
1
1
2
1
2
1
1

12
= −1  −12
0


1
2
1
2
1
2

8
= −1  −12
8


1
2
1
2
1
2

5
= −1  −10
5


1
2
2
1
1
1

12
= −1  −8
−8


1
2
2
1
2
2

= −1 
5
0
−5



2
1 −2
1
2  =  −2 −2
1 
2
1
3
1


−5
0
6
10
0 
+ 0 6
/15
−5
0
−9


2
−3
1
1  =  2 −3
1
2
1


0 −12
0
8 
+ 0
/20
0
8
−1
0
0
0
0
0
−6
−6 
+ 5
/15
9

1
−1
1  0
1
0
0
0
0
/20
−5
−5 
+ 4
/20
15
−1
0
0
0
0
0
/20
−5
5
15 −15 
+ 4
/20
−5
5



2
−1 −1
4
−1
2  =  0
1
6  0
1
1
0
5
0


5 −10
6 −9
6
0
0 
9 −6
+ 0  −6
/15
−5
10
0
0
0
/15
12
−5
1

−8
15 
/20
1

7
7
8
8 
/20
5
5
8
−5
1

8
15 
/20
1

8
8
7
7 
/20
5
5
−4
0
8
8
8
8

0
5 
/20
5

5
5 
/20
5
5
5
5

0
3 
/15
6

6
6 
/15
6
5
5
5

4
5 
/20
7

7
7 
/20
7
7
8
5
8
7
5
7
7
7
−5
0
5
4
4
4
0
−5
5
8
8
8
−5
9
1
4
6
5

10
−6 
/15
1

4
4
6
6 
/15
5
5
60

Functions of matrices


1
−3 −1

1  =  2 −1
1
2
3



12 −12
0
8
0 
= −1  −8
+ 0
/20
−8
8
0

1
−1
1  0
1
0
0
0
0


0
−4
0  0
4
8



1
−1
1  0
1
0
0
0
0


0
−5
0  0
5
5




1
−1 −1
4
−1

2  =  1
0
5  0
2
0
1
6
0



5 −10
5
6
6 −9
10 −5 
0
0
= −1  −5
+ 0 0
/15
0
0
0
−6 −6
9
0
0
0


0
−5
0   −6
5
1


+ 5




2
−1
0
5
−1
1  =  1 −1
4  0
2
0
1
6
0


−5 −5
0
0
0
5
5 
6 −9
+ 0 6
/15
0
0
−6 −6
9
0
0
0


0
−10
0   −6
5
1


+ 5



2
−1
0
5
−1
2  =  0 −1
6  0
1
1
1
4
0


−5 −5
0
0
0
0
0 
9 −6
+ 0  −6
/15
5
5
6 −9
6
0
0
0


0
−10
0  6
5
1


+ 5


2
−2
2
1  =  1 −3
1
1
2


0 −10
0
5 
+ 0
/15
0
5
0
0
0

1
2
2
2
1
1


2
−2 −2

2  =  1 −2
2
1
3



10 −10
0
5
0 
= −1  −5
+ 0
/15
−5
5
0


1
2
2
2
1
1
1
2
2
1
2
2
2
1
2
10
= −1  −10
0

1
2
2
2
2
1

= −1 


0
0
0
6
6
−9
−5
−5 
+ 4
/20
15
−6
−6 
+ 5
/15
9
2
1
2


0
5
0
5
0 −15
1
2
2
10
0
−10
2
2
2

10
= −1  −5
−5

1
−1
1  0
1
0
0
0
0
−6
9
−6
/15
/15
/15

0
−5
0  0
5
5

6
−9 
+ 5
/15
6
7
7
7

0
5 
/20
5

5
5 
/20
5
4
4
4

0
3 
/15
6

6
6 
/15
6
4
−5
7
8
8
8
5
−3
4
5
5
5
10
−6
1

−5
9 
/15
1

4
4
5
5 
/15
6
6
5
−6
1
5
4
6

5
9 
/15
1

5
4 
/15
6
5
6
4

5
6 
/15
1

5
6 
/15
4
6
6
6

5
3 
/15
4

4
4 
/15
4
4
5
6
5
4
6
5
−9
1
5
6
4
0
−3
6
5
5
5
1.6. 3 × 3 singular matrices


2
1
1
= 0





1
1 −2
1
2  =  −3
1
1 
2
1
1
1


−3
3
8
0
9 −9 
0
+ 1  −4
/12
−3
3
−4
0
0
0
0



1
0 −1
4
1  =  −1
0
3 
2
1
1
5


0
0
8 −4
9 −3 
0
+ 1 0
/12
−9
3
−8
4
0
0
0



1
0 −1
4
2  =  −1
1
5 
1
1
0
3


0
0
8 −4
3 −9 
4
+ 1  −8
/12
−3
9
0
0
0
0
0



2
−1 −1
5
1  =  0
1
4 
1
1
0
3


3 −9
4 −8
0
0 
8
+ 1  −4
/12
−3
9
0
0
0
0
0



1
−1 −2
1
1  =  −1
1
1 
1
3
1
1


3 −3
8 −8
3 −3 
4
+ 1  −4
/12
−9
9
−4
4
0
0
0



2
−2 −3
1
2  =  −2
1
1 
2
3
1
1


0
8 −8
15 −15
0
8 −8 
5
+ 1  −5
/20
0 −12
12
−5
5
0
0
0
1
1
1

2
1
1
0
0
0
1
1
2

0
= 0  −3
3


2
1
1
1
2
1

= 0


2
1
1
= 0


2
1
1
= 0

3
0
−3
1
2
2


0
3
−3
1
2
1

2
1
1
0
0
0
1
2
2

= 0
61

0
0
0   −4
4
4


−8
4
4 
+ 4 4
/12
4
4
−3
0
3

0
3
0   −8
4
1


−4
4
0 
+ 4 3
/12
4
5
−9
4
1

0
−3
0   −8
4
1


−4
4
4 
+ 4 5
/12
0
3
−3
4
1

0
−3
0   −4
4
1


4
5
−4 
+ 4 4
/12
0
3
−3
8
1

0
0
0   −4
4
4


0
4
0 
+ 4 4
/12
0
4
−3
4
5

0
0
0   −5
5
5


0
5
0 
+ 5 5
/20
0
5
−4
5
7
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
0
1
0
3
3
3
4
3
5
4
5
3
5
4
3
5
5
5
7
7
7

3
4 
/12
5

5
5 
/12
5

3
4 
/12
1

4
3 
/12
5

9
4 
/12
1

4
5 
/12
3

9
−4 
/12
1

5
4 
/12
3

3
0 
/12
3

3
3 
/12
3

4
0 
/20
8

8
8 
/20
8
62


Functions of matrices
2
1
1
2
1
1

3
= 0  −3
0


2
1
1
2
2
2

= 0


2
1
2
1
2
1

= 0


2
1
2
= 0

2
1
2
= 0

8
8
−12
1
2
2


3
3
−9
1
2
1


0
0
0
2
1
2
0
8
−8
1
2
2

= 0
8
0
−8



1
−1 −1
5
1  =  1
0
3 
2
0
1
4


−9
3
4
4
9 −3 
0
+ 1 0
/12
0
0
−4 −4

0
−3
0   −4
4
1


−8
5
0 
+ 4 3
/12
8
4
9
−4
1




1
2 −3
1
0
0
0
0
2  =  −3
1
1  0
1
0   −5
2
2
1
1
0
0
5
5




−8
8
15
0 −15
5
12 −12 
0
5 
+ 1  −5
+ 5 5
/20
/20
−8
8
−5
0
5
5
−4
0
8


1
−1
1
1
1  =  −1 −2
1
1
3
1
1


0 −3
4
0 −3 
+ 1  −8
/12
0
9
4



2
−2
1
1
2  =  −2 −3
1
2
3
1
1


0 −8
5
0 −8 
+ 1  −15
/20
0
12
5


0
0
0
5
3
4
8
8
8

0
−3
0  4
4
5


0
5
0 
+ 4 5
/12
0
5
0
−4
4

0
−4
0  5
5
7


0
7
0 
+ 5 7
/20
0
7
0
−5
5

0
−8
0   −15
5
1


−5
5
5 
+ 5 7
/20
0
8
−8
5
1

0
−8
0   −5
5
1


5
7
−5 
+ 5 5
/20
0
8
−8
15
1
0
0
0
−4
8
−4

0
1
0
0
0
0
−5
15
−5



1
0 −1
5
2  =  −1
1
7 
2
1
0
8


0
0
15 −5
8 −12 
5
+ 1  −15
/20
−8
12
0
0
0
0
0



2
−1 −1
7
1  =  0
1
5 
2
1
0
8


8 −12
5 −15
0
0 
15
+ 1  −5
/20
−8
12
0
0
0
0
0
0
1
0
0
1
0
0
1
0
0
1
0
4
4
4
5
5
5
5
7
8
7
5
8

−3
8 
/12
1

5
3 
/12
4

4
5 
/20
7

7
7 
/20
7

3
0 
/12
3

3
3 
/12
3

4
0 
/20
8

8
8 
/20
8

12
5 
/20
1

5
7 
/20
8

12
−5 
/20
1

7
5 
/20
8
1.6. 3 × 3 singular matrices


2
1
2
2
1
2

= −1 

0
5
−5
63



2
0 −1
6
−1
2  =  −1
1
4  0
1
1
0
5
0


0
0
9 −6 −6
5 −10 
6
6
+ 0  −9
/15
−5
10
0
0
0



2
−1
0
8
1  =  0 −1
5 
2
1
1
7


−8 −8
0
0
0
0 
15
+ 1  −5
/20
8
8
5 −15
0
0
0



1
1 −1
1
1  =  −3 −1
1 
2
1
2
1


−3
0
4
0
9
0 
0
+ 1 4
/12
−3
0
−8
0
0
0
0



1
0 −1
5

2  =  −1
0
8 
2
1
1
7



0
0
0
15 −5
12 −8 
0
= 0  −8
+ 1 0
/20
8 −12
8
−15
5
0
0
0

2
1
2
2
2
1

= 0


2
2
1
12
0
−12
1
1
1

3
= 0  −9
3

2
2
1

2
2
1

2
2
1
1
2
2

0
−5
0   −9
5
1


+ 5
/15
6
4
5

10
6 
/15
1

6
6
4
4 
/15
5
5

8
5 
/20
1

8
5 
/20
7

0
3
0   −4
4
5


−4
5
−4 
+ 4 5
/12
8
5
3
3
3

0
4 
/12
4

4
4 
/12
4
0
1
0

0
8 −12
0   −15
5
5
1
1


−5
5
5
0 
8
+ 5 8
/20
5
7
7

8
5 
/20
1

5
8 
/20
7

0
5 −10
0   −9
6
5
1
1

6
6

5
+ 5 5
/15
4
4

5
6 
/15
1

6
5 
/15
4
0
1
0
0
1
0
−3
0
3
0
0
0





1
−1 −1
7
0
0
0
−8

2  =  1
0
8  0
1
0   −5
2
0
1
5
0
0
5
1





8 −12
8
5
5 −15
7
12 −8 
0
0 
= 0  −8
+ 1 0
+ 5 8
/20
/20
0
0
0
−5 −5
15
5
2
2
1
−5
6
1

0
−12
8
0   5 −15
5
1
1


0
8
8
−5 
5
+ 5 5
/20
5
7
7



2
0 −1
6
−1

2  =  −1
0
5  0
1
1
1
4
0



0
0
0
9 −6 −6
10 −5 
0
0
= −1  −5
+ 0 0
/15
5 −10
5
−9
6
6
2
1
2
0
0
0

12
−5
1
7
8
5

−8
15 
/20
1

7
8 
/20
5
64


Functions of matrices
2
2
1
2
2
1

12
= 0  −12
0


2
2
1
2
2
2

8
= 0  −12
8


2
2
2
1
1
2

= −1 


2
2
2
12
= 0  −8
−8

2
1
2

0
0
0
= −1 




1
2 −1
1
1  =  −3 −1
1 
2
2
3
1


−8
0
5
0
12
0 
0
+ 1 5
/20
−8
0
−15
0
−4
0
8


2
−3
1
1  =  2 −3
2
2
1


0 −12
0
8 
+ 1
/20
0
8
1
2
1
2
2
2
8
−5
1
2
2
1
0
0
0

0
0
0   −3
5
6

0
0 
+ 5
/15
0

1
1 
1
0
5
0
5
0 −15
0
0
0
−1
0
0
−5
3
4
6
6
6
0
1
0
0
0
0


0
0
0   −3
5
6

−9
6 
+ 5
/15
6
0
0
0

0
−4
0  0
5
8


−5
8
−5 
+ 5 8
/20
15
8
0
0
0
0
1
0

0
5 
/20
5

5
5 
/20
5
8
8
8

0
−4
0  0
5
8


−5
5
8
15 −15 
+ 5 8
/20
−5
5
8



1
1 −3
1
2  =  −2
2
1 
1
1
2
1


−5
5
9
10 −10 
+ 0  −6
/15
−5
5
−6


1
−3 −1

1  =  2 −1
2
2
3



12 −12
0
8
0 
= 0  −8
+ 1
/20
−8
8
0
2
2
2

1
1 
1
0
1
0
0
0
0


8
15 
/20
1

8
7 
/20
5
8
7
5

0
4
0   −5
5
7


−5
7
−5 
+ 5 7
/20
15
7
0
0
0



2
−1 −3
1
−1
2  =  −1
2
1  0
1
2
2
1
0


0
5 −5
9 −9
0
5 −5 
6
+ 0  −6
/15
0 −10
10
−6
6






2
−1
0
8
0
0
0
−12
1  =  1 −1
7  0
1
0   −5
2
0
1
5
0
0
5
1




−8 −8
0
0
0
8
8
8 
5 −15 
+ 1 5
+ 5 7
/20
/20
0
0
−5 −5
15
5
4
4
4

4
5 
/20
7

7
7 
/20
7
0
−5
5
5
5
5
−5
0
5
6
6
6
5
5
5
4
−5
7
7
7
7

5
0 
/15
5

5
5 
/15
5

5
3 
/15
4

4
4 
/15
4

0
5 
/20
5

5
5 
/20
5
1.6. 3 × 3 singular matrices



1
−1
1
1
−1  =  1 −1
0
1
0
1
1


1
0 −1
−1
0
1  + 1 1
= 0  −1
0
0
0
−1
1
1
1


1
1
1



1
−1 −1

−1  =  1
1
−1
1
0



1
1 −3
3 
= −1  −1 −1
+ 0
/3
−1 −1
3

65
1
1
1
1
1
1



0
−1
0
1
0   −1 −1
1 
2
1
1
0



1
1
1
0
−1  + 2  0
0
0 
1
1
1
0
0
0
0

−1
1
−1

2
−1
1  0
1
0
−3
3
0
0
0
0

0
−1
0  0
2
1

3
2
−3 
+ 2 1
/3
0
1
−1
3
1

0
−1
0   −5
3
2

−1
5
2
0
0
0




2
−1 −1
2
−2

2  =  −1
1
2  0
−1
2
0
1
0



1
1 −4
5 −5
1 −4 
5
= −2  1
+ 0  −5
/10
−2 −2
8
0
0
1
1
1

1
1
1


2
−1
1
1

−1  =  1 −2
0
2
0
1
1



2
0 −2
−1
0
2 
= 0  −2
+ 1 2
/2
0
0
0
−1
1
1
1

1
1
1
1
1
1

1
1
1

= 0

0
1
0




−1
−1 −1
0
1  =  1
1
1
1
0
1
1


0 −1
1
1
0
1 −1  + 1  −1
0
0
0
−1
−1
2
−1


1
−1
−1
0
0
0

0
0 
+ 3
/10
0
0
1
0
1
1
1
4
4
2
0
−1
1
1
0
1

4
0 
/10
2

4
4 
/10
2

2
1 
/2
1

1
0 
/2
1


0
0
1 −1
0   −1 −1
1 
2
1
1
0



−1
0
0
0
1  + 2 1
1
0 
1
1
1
0
0
0
0



−1
1 −1
1
−1

1  =  −1
1
2  0
−1
1
0
1
0



−1 −1
3
3
0 −3
1 −3 
0
3
= −1  1
+ 0  −3
/3
−1 −1
3
0
0
0
1
1
1
4
4
2

0
−2
0   −1
3
1


1
1
−2 
+ 3 0
/2
1
1
0
0
0
2
1
1

3
−3 
/3
0

0
0 
/3
0
0
1
0

0
−1
0   −3
2
1

1

+ 2 2
/3
1
0
0
0

−1
0
1
1
2
1

3
3 
/3
0

0
0 
/3
0
66
Functions of matrices

1
1
1

1
1
1

= 0





−1
−1 −2
0
2  =  1
1
1
2
0
1
1


0 −2
2
2
0
2 −2 
+ 1  −1
/2
0
0
0
−1


2
−1
−1

0
0
0   −1
3
1


−2
0
1 
+ 3 1
/2
1
1
0
0
0



1
−1 −1
1
1  =  0 −1
1 
1
1
1
0


1 −1
0
−1
1
0
0  + 1  −1
1
= 0 0
−1
1
0
1 −1
1
1
1
0
1
0

0
0
0
0
1
0

1
1
−1

= 0

1
1
0

1
0 
/3
1

1
1 
/3
0

0
−1
1
0   2 −2
2
3 −1


−4
3 −1
−4 
+ 2  3 −1
/2
2
0
0

0
2 
/2
4

4
4 
/2
0
1
1
−1

1
1
1
3
1
−2



2
−3 −2
1
2  =  −1 −2
1 
1
2
1
0


−3
0
−4
4
−1
0 
4
+ 1  −4
/2
2
0
2 −2
0
0
0


1
1
−1

1
1
1
1
1
−1



−1
−1  = 
1
0
= 0  −1
−1
0
1
1
0
1
1
1
1
1


0
0  + 1
0

1
1 
0
1
1
1
−1
−1
−1
1
−3
1
0
1
0




2
−2 −1
1
−1
0
0
−2

2  =  −2 −3
1  0
0
0  3
−1
3
2
0
0
0
2
5




4 −4 −4
−3
3
0
5
9
0 
= −1  4 −4 −4 
+ 0  −9
+ 2 5
/6
/6
−6
6
6
6 −6
0
0
1
1
1
0
1
1

−2
1 
/2
1

0
1 
/2
1


0
−1
1
0
0   1 −1
1 
2
1
0
1



−1
1
0
1
−1  + 2  1
0
1 
1
0
0
0
1
1
−1





1
−1
0
1
−1
0
0
−1

1  =  −1 −1
1  0
0
0  3
−1
3
1
0
0
0
2
2




1 −1 −1
0
0
0
2
3
0 
= −1  1 −1 −1 
+ 0  −3
+ 2 2
/3
/3
−3
3
3
3 −3
0
0
1
1
1
2
−1
1
0
0
0
2
−3
1
1
1
0

2
0 
/6
4

4
4 
/6
0


0
−1
1
0
0   1 −1
1 
2
0
1 −1



1
0
1 −1
1  + 2 0
1 −1 
1
0
0
0
0
1
0
1.6. 3 × 3 singular matrices

1 −1

1 −1
−1 −1

−1
= −1  −1
−3

1
1
1


 = 
1
1
3
67
1
1
3
1
0
1


1
1 
+ 0
/3
3

1
−1
1  0
0
0
3
0
3


2
−5
1
1

−1  =  3 −1
0
2
1
1
1



5
0 −5
−3
0
3 
= 0  −3
+ 2 3
/6
−1
0
1
−3

1
1
1
1
2
1



−6
6
−6



−1
3 −2
0
1  =  −2
1
1
1
1
1
1


0 −6
6
6
0
4 −4 
+ 1  −3
/6
0 −2
2
−3



−1
4 −3
0
2  =  −3
1
1
2
1
1
1


0 −12
12
12
0
9 −9 
+ 1  −4
/12
0 −3
3
−4

1
2
2

= 0


1
1
1
1
2
2

= 0


0
−1
0   −3
3
4


3
4
−3 
+ 3 0
/6
3
4
0
−6
6

0
−6
0  4
3
2

−3
0
3
0
0
0
−1
0
0
1
2
1
1
1
1
1
−3
2
−3
0
−3



−1
1
4
1

2  =  −1 −3
3 
−1
1
1
1



−6 −3
15
16
3 −15 
= −1  6
+ 0  −12
/12
−6 −3
15
4
1
1
1

0
−1
0  3
2
1

0
1
0 
+ 2 1
/3
0
0
1
−1
1

0
2
0
0
0
0

0 −16
0
12 
+ 3
/12
0 −4
2
2
0
6
0
6
2
6
2
−2
−3
5

0
0

0   −4
4
4


12 −12
0
−4
4 
+ 4 4
/12
−4
4
4
−3
−4
7

6
−3
−3
0
0
0
0
1
0
0
1
0

0
1
0   −3
2
2

2

+ 2 0
/3
2
0
0
0


1
0 
/3
−1

−1
−1 
/3
0

1
3 
/6
2

2
0 
/6
2

15
−4 
/12
1

3
1
9
3 
/12
3
1

0
0
0   −3
3
3


−6
0
3 
+ 3 3
/6
3
3
0
0
0



1
1
0
1
−1

−1  =  −3 −1
0  0
1
1
1
1
0


1 −1 −1
0
0
0
3
3 
0 −3
= −1  −3
+ 0 3
/3
1 −1 −1
−3
0
3
1
1
1
0
0
0

0
5
5
0
7
7
−1
0
1
1
0
1

2
3 
/6
1

0
1 
/6
1

3
4 
/12
5

0
5 
/12
5

−1
3 
/3
1

1
0 
/3
1
68
Functions of matrices




2
1 −1
1

−1  =  −4 −3
0 
2
1
2
1


3 −3 −3
4
12
12 
= −1  −12
+ 0  12
/12
3 −3 −3
−8
1
1
1


1
1
1
1
−1
1

1
−1
1

= −2 


−1
1
1  =  −2
−1
1

0 −1
1
0
2 −2 
+
/3
0 −1
1
−1
0
0

0
0  −3
−3
0
0
0
1
−1
2




2
0 −2
6
−2

2  =  −2
0
4  0
2
1
1
7
0


0
0
0
36 −12 −24
40 −16 
0
0
= −2  −8
+ 0 0
/48
4 −20
8
−18
6
12
0
0
0


1
1
1
1
−1
−1

0
0
0


1
= −1  −1
−1



−1
0

−1  =  1
−1
1

0
0
0
2
1 
= −2  −1
+
/3
−1
2
1
1
1
1
1
−1
−1

0
0
0
3
0
−3

3
−2
1  0
1
0
1
0
1

0
0
0
0
−3
0
−3
5
0
5

0
0
0   −3
1
3

0
3
3 
+ 1 3
/3
3
3
1
1
1

2
−1
1  0
1
0
−3
0
3
0
0
0




1
−1 −1
1  =  1
0
1
1
1


−3
1
3 −1 
+ 0
/3
3 −1

0
3
0   −4
3
5

0 −4
0 −12 
+ 3
/12
0
8

1
−2
1  0
1
0
0
1
1
0
0
0


−3
4 
/12
7

3
7
0
0 
/12
3
7
−1
0
1
1
1
1

1
3 
/3
−1

−1
−1 
/3
−1


0
4 −20
8
0   −18
6
12 
/48
4
2
2
4


12
12
24

8
16 
+ 4 8
/48
/48
14
14
28

0
−1
0  0
2
1

−3
2
0 
+ 2 1
/3
3
1
3
−3
0

0
−1
0  0
1
1

3
3
0 
+ 1 1
/3
3
1
2
−3
1
0
0
0

0
0
0
3
1
1

−1
3 
/3
1

2
1 
/3
1

1
3 
/3
−1

−3
−1 
/3
−1
1.7. 3 × 3 symmetric matrices
1.7
69
3 × 3 symmetric matrices
The following 3 × 3 matrices have nonzero integer entries, are nonsingular, have distinct integer eigenvalues, and are symmetric:






1
1
2
−1
1
1
−1
0
0
−3
0
3
 1
2
1  =  0 −2
1  0
1
0   1 −2
1 
/6
2
1
1
1
1
1
0
0
4
2
2
2






3
0 −3
1 −2
1
2
2
2
4 −2 
2
2 
0
0 
= −1  0
+ 1  −2
+ 4 2
/6
/6
/6
1 −2
1
2
2
2
−3
0
3


1
2
1
2
1
1

3
= −1  −3
0


1
2
2
2
2
1

4

= −1 −2
−2


2
1
1
1
1
2

0
0
0
= −1 


2
1
2
1
2
2

= −1 


2
2
1
1
1
−2
2
1
2

1
= −1  −2
1



1
−1 −1
1
−1




1
1 −1
1
0
=
2
0
2
1
0


−3
0
1
1 −2
3
0 
1 −2
+ 1 1
/6
0
0
−2 −2
4
0
1
0


0
−3


0
−1
4
2

2

+ 4 2
/6
2
3
−1
2


2
−2
0
1  =  1 −1
2
1
1


−2 −2
1
1 
+ 1
/6
1
1
0
1
0


0
−2
0  0
5
2

0
2


−3
2
+ 5
/6
3
2
1
−3
2

0
0
0   −2
4
2

2

+ 4 2
/6
2
−3
1
2




2
−1 −1
1
−1
0
0
−1
2  =  −1
1
1  0
1
0   −3
1
2
0
1
0
0
5
2




3 −3
0
2
1 −2
3
0 
1 −2 
+ 1  −3
+ 5 2
/6
/6
0
0
0
2
−2
4
−1
3
2



1
1 −1
1
−1
2  =  −2
0
1  0
2
1
1
1
0


−2
1
3
0 −3
4 −2 
0
0
+ 1 0
/6
−2
1
−3
0
3
−2
0
2

1
−1
1  0
1
0
0
0
0
0
3
−3



1
0 −2
1
−1
2  =  −1
1
1  0
1
1
1
1
0


4 −2 −2
0
0
1
1
3 −3 
+ 1  −2
/6
−2
1
1
−3
3
0
1
0


0
1
0   −3
5
2

2

+ 5 2
/6
2
0
1
0

2
2
2
2
2
2
2
2
2
2
2
2
2
2
2

0
2 
/6
2

2
2 
/6
2

1
3 
/6
2

2
2 
/6
2

3
1 
/6
2

2
2 
/6
2

2
0 
/6
2

2
2 
/6
2

1
3 
/6
2

2
2 
/6
2
70
Functions of matrices




1
−1
1
1
−2

−1  =  1 −1
1  0
−1
2
1
0
0


1 −1 −2
2 −2
2
1
2 
2 −2
= −2  −1
+ 1  −2
/6
−2
2
4
2 −2
2


1
1
1
1
1
1
1
−1
1

= −2 


0
−1
0  2
2
3

3

+ 2 3
/6
0
1
−2
3




1
0 −1
2
−2
0
0
0
1  =  −1
1
1   0 −1
0   −2
−1
1
1
1
0
0
2
2




0
0
0
2 −2 −2
4
0
3 −3 
2
2 
− 1  −2
+ 2 2
/6
/6
0 −3
3
−2
2
2
2
−3
2
1
1
1
−1

0
1
0

3
3
0
2
1
1



1
−1 −1
1
−2

−1  =  2 −1
0  0
1
1
1
1
0


1 −2 −1
2
2 −2
4
2 
2 −2
= −2  −2
+ 1 2
/6
−1
2
1
−2 −2
2
0
1
0


0
−1
0   −2
2
3

3

+ 2 0
/6
3
2
−2
0

0
1
0


0
1
0   −2
2
3

3

+ 2 3
/6
0
−1
2
3

0
2
0   −2
4
0

0

+ 4 0
/6
0
−2
−1
3

0
1
0  2
2
−3

2
3
2 
+ 2 0
/6
2
−3
−2
2
0
1
1
1
1
−1
−1




−1
1 −1
1
−2
1  =  −1
1
1  0
−1
2
1
0
0



1 −1
2
2 −2 −2
1 −2 
2
2
= −2  −1
+ 1  −2
/6
2 −2
4
−2
2
2
1
 1
−1

1
1
1



−1
1 −2
0
−1
2  =  −1 −1
1  0
2
1
1
1
0



2 −2
2
4
2 −2
2 −2 
1 −1
= −1  −2
+ 2 2
/6
2 −2
2
−2 −1
1
1
 1
−1

1
2
2


−1
1
1  =  −2
1
1

1 −2
1
4 −2 
= −2  −2
+
/6
1 −2
1
1
 1
−1
1
−1
1


−1
−2
0  0
1
0
1
1
1

1
2
2
2
2
2
2
0
2
0

0
1
0

0
0
0
3
3
0
0
3
3
0
0
0

2
2 
/6
0

0
0 
/6
0

3
2 
/6
1

2
1 
/6
1

1
2 
/6
3

3
0 
/6
3

2
2 
/6
0

0
0 
/6
0

2
1 
/6
3

0
3 
/6
3

1
2 
/6
3

−3
0 
/6
3
1.7. 3 × 3 symmetric matrices

1
 1
−1
1
−1
−1

= −2 





−1
0
1 −2
−2
0
0
0
−1  =  1 −1 −1   0 −1
0  2
−1
1
1
1
0
0
2
−2




0
0
0
2 −2
2
4
0
3
3 
2 −2 
− 1  −2
+ 2 2
/6
/6
0
3
3
2 −2
2
−2


−1
−1
−1  =  1
2
0

3 −3
0
3
0 
= −1  −3
+
/6
0
0
0
1
 2
−1
2
1
−1


71

−1
−1
−1   0
1
0
1
1
2

1
1
1
2
1
1
2

0
−3
0  1
4
−2

2
2
2 
+ 4 2
/6
4
−2

3
1
−2

0
2
0   −3
4
1

1

+ 4 2
/6
1
−2
0
2
0
1
0


0
−1
0  2
2
−3

2
3
2 
+ 2  −3
/6
2
0
−1
2
3
2
2
2


1
−1
1  =  −1
−1
2

1
1 −2
1 −2 
= −2  1
+
/6
−2 −2
4
1
 −1
1

1
 −1
1
−1
1
1

−1
2
2

= −1 

1
 −1
1
2
2
−2
−1
−1
1

= −2 
1
2
−1

−1
−2
1  0
0
0
1
1
1

1
2
2
2
2
2
2
2
1
−1
0
1
0




−1
1 −1
1
−2
2  =  −1
0
2  0
1
1
1
1
0



2 −2
2
3
0 −3
2 −2 
0
0
= −2  −2
+ 2 0
/6
2 −2
2
−3
0
3
1
 2
−1
3
−2
−1
0
2
0

2
2
−2
2
4
2
−3
3
0



1
−1
2
0
−1
2  =  −1 −1
1  0
2
1
1
1
0


2 −2
4 −2
2
2 −2 
1 −1
+ 2  −2
/6
−2
2
2 −1
1
0
2
0


0
−2
0  2
4
0

0

+ 4 0
/6
0
−2
−1
3



1
−1 −1
1
−2
1  =  −2
1
0  0
1
1
1
1
0


2 −1
2 −2 −2
4 −2 
2
2
+ 1  −2
/6
−2
1
−2
2
2
0
1
0


0
−1
0   −2
2
3

3

+ 2 0
/6
3
−2
2
0
0
3
3
0
0
0

3
2 
/6
1

−2
−1 
/6
1

0
2 
/6
2

−2
−2 
/6
2

2
3 
/6
1

1
2 
/6
1

2
2 
/6
0

0
0 
/6
0

2
1 
/6
3

0
3 
/6
3

1
2 
/6
3

3
0 
/6
3
72
Functions of matrices

1
 −1
1
−1
−1
−1

= −2 

1
 −1
2
−1
1
2

= −2 





1
0 −1
2
−2
0
0
0
−1  =  1 −1 −1   0 −1
0   −2
−1
1
1
1
0
0
2
2




0
0
0
2
2 −2
4
0
3
3 
2 −2 
− 1 2
+ 2  −2
/6
/6
0
3
3
−2 −2
2
2
3
−2
−1




2
−1 −1
1
−2
0
0
−2
2  =  −1
1
1  0
2
0   −3
2
1
0
2
0
0
4
1




2 −2
3 −3
0
1
2 −2 
3
0 
+ 2  −3
+ 4 1
/6
/6
−2
2
0
0
0
2
−2
3
1
2
2
−2


2
−1
−1  =  0
1
1

3
0 −3
0
0 
= −1  0
+
/6
−3
0
3
1
 −1
2

1
 −1
−1
−1
2
−1

−1
1
−1

= −2 

1
 −1
−1
−1
−1
1

= −2 

1
 −1
−1
−1
−1
−1

= −2 

1
−1
−1   0
1
0
1
2
1

1
1
2
1
2
4
2

0
−3
0  1
4
2

1
2
2 
+ 4  −2
/6
1
2

−2
−2
1  0
1
0
1
1
1

1
2
2
2
2
2
2
1
1
2
0
1
0

0
2
−2

0
1
0   −2
2
−3

3

+ 2  −3
/6
0
1
−2
3

0
0
0
−1
0  2
0
2
−2


2
4
2 
+ 2  −2
/6
2
−2
−3
2
1



−1
1 −1 −1
−2
−1  =  1 −1
1  0
−1
2
1
0
0


1
1
2
2
2 −2
1
1
2 
2 −2
+ 1 2
/6
2
2
4
−2 −2
2


−1
0
1  =  −1
−1
1

0
0
0
0
3 −3 
−
/6
0 −3
3
−2
1
−1



−1
1
1 −1
−2
−1  =  2 −1
0  0
1
1
1
1
0


1
2
1
2 −2
2
2
4
2 
2 −2
+ 1  −2
/6
1
2
1
2 −2
2
0
1
0


0
1
0  2
2
−3

3

+ 2 0
/6
−3
0
1
0

−2
2
−2
−3
3
0
−2
1
1
2
−2
0
0
0
0

3
2 
/6
1

2
−1 
/6
1

2
0 
/6
2

2
2 
/6
4

3
1 
/6
2

2
−2 
/6
2

2
2 
/6
0

0
0 
/6
0

3
2 
/6
1

−2
1 
/6
1

1
2 
/6
3

−3
0 
/6
3
1.7. 3 × 3 symmetric matrices

73



2
−1 −1
1
−1

−1  =  1 −2
0  0
2
1
1
1
0


2 −2 −2
1
2 −1
2
2 
4 −2
= −1  −2
+ 2 2
/6
−2
2
2
−1 −2
1
0
2
0


0
−2
0   −1
4
3

3

+ 4 0
/6
3
2
−2
0




2
−1 −1
1
−2

−1  =  2 −1
0  0
2
1
1
1
0


1 −2 −1
2
2 −2
4
2 
2 −2
= −2  −2
+ 1 2
/6
−1
2
1
−2 −2
2
0
1
0


0
−1
0   −2
4
3

3

+ 4 0
/6
3
2
−2
0




1
−1
1
1
−1

−1  =  1 −1
1  0
1
1
2
0
0


2 −2 −2
1 −1
2
2
2 
1 −2
= −1  −2
+ 2  −1
/6
−2
2
2
2 −2
4
0
2
0


0
−2
0  1
4
3

3

+ 4 3
/6
0
2
−1
3




1
−1
1
1
−2

−1  =  1 −1
1  0
−1
2
1
0
0


1 −1 −2
2 −2
2
1
2 
2 −2
= −2  −1
+ 1  −2
/6
−2
2
4
2 −2
2
0
1
0


0
−1
0  2
4
3

3

+ 4 3
/6
0
1
−2
3

0
2
0


0
−2
0  0
4
2

0
4
−3 
+ 4 2
/6
3
2
2
−3
1

0
2
0   −1
4
3

3

+ 4 3
/6
0
−2
1
3
2
1
2
2
1
2
2
2
1
2
2
1
1
1
−1

1
−1
−1

2
2
−1

2
2
−1



2
−1
0

−1  =  1 −1
1
1
1


2 −2 −2
2
2 
= −2  −2
+ 2
/6
−2
2
2

2
2
2
2
1
−1


2
−2
1  0
1
0
0
0
0
0
3
−3



−1
1 −1
1
−1
1  =  −1
1
1  0
1
1
2
0
0



2 −2
2
1 −1 −2
2 −2 
1
2
= −1  −2
+ 2  −1
/6
2 −2
2
−2
2
4
2
 2
−1
2
2
1
0
2
0

0
0
0
0
0
0
3
3
0
3
3
0
2
1
1
3
3
0

2
1 
/6
3

3
0 
/6
3

1
2 
/6
3

3
0 
/6
3

2
2 
/6
0

0
0 
/6
0

2
2 
/6
0

0
0 
/6
0

2
3 
/6
1

2
1 
/6
1

2
2 
/6
0

0
0 
/6
0
74
Functions of matrices




−1
1 −1
1
−2
1  =  −1
1
1  0
−1
2
1
0
0



1 −1
2
2 −2 −2
1 −2 
2
2
= −2  −1
+ 1  −2
/6
2 −2
4
−2
2
2
2
 2
−1

2
2
1
0
1
0


0
1
0   −2
4
3

3

+ 4 3
/6
0
−1
2
3
3
3
0



2
−1 −1
1
−1
1  =  −1
2
0  0
2
1
1
1
0


2 −2
1 −2 −1
2 −2 
4
2
+ 2  −2
/6
−2
2
−1
2
1
0
2
0


0
−2
0   −1
4
3

3

+ 4 0
/6
3
−2
2
0



2
−1 −1
1
−2
1  =  −2
1
0  0
2
1
1
1
0


2 −1
2 −2 −2
4 −2 
2
2
+ 1  −2
/6
−2
1
−2
2
2
0
1
0


0
−1
0   −2
4
3

3

+ 4 0
/6
3
−2
2
0

0
0
0  2
4
−2

2
2
1 
+ 4  −2
/6
1
−2
−3
1
2




1
−1
1
1
−2
0
0
−3
1  =  0 −1
2   0 −1
0  2
−1
1
1
1
0
0
2
1





3
0 −3
2 −2
2
1
0
0 
2 −2 
= −2  0
− 1  −2
+ 2 2
/6
/6
−3
0
3
2 −2
2
1
0
−2
2
2
 −1
2
−1
1
1

= −1 

2
 −1
2
−1
−1
1

= −2 

2
 −1
−1
2
2
−2
1
2
−1
−1
1
2

= −1 

−1
 1
1



−1
0
2  =  −1
1
1

0
0
0
0
3 −3 
+
/6
0 −3
3

1
4
2
2
2
1
1
0
1
0

1
1
1


1
−2
−1  =  1
1
1

4 −2 −2
1
1 
= −2  −2
+
/6
−2
1
1
−1
 1
1

−1
−1
1  0
1
0
2
1
1
1
1
−1


0
−2
−1   0
1
0
1
1
1

1
2
2
2
2
2
2

0
−2
0  2
2
0

2
0
2 
+ 2 0
/6
2
0
0
1
0

0
0
0
0
0
0
−2
2
2
2
4
2
1
2
−3
0
3
−3

2
2 
/6
0

0
0 
/6
0

2
1 
/6
3

3
0 
/6
3

1
2 
/6
3

3
0 
/6
3

3
1 
/6
2

−2
2 
/6
2

3
2 
/6
1

1
2 
/6
1

1
2 
/6
3

0
−3 
/6
3
1.7. 3 × 3 symmetric matrices

−1
 1
1
1
−1
1

3
= −2  −3
0

75




1
−1 −1
1
−2
0
0
−3
1  =  1 −1
1   0 −1
0   −2
1
0
1
2
0
0
2
1




−3
0
2
2 −2
1
3
0 
2 −2 
− 1 2
+ 2 1
/6
/6
0
0
−2 −2
2
2



−1
2 −1
0
−2
1  =  −1 −1
1  0
1
1
1
1
0



4 −2
2
2
2 −2
1 −1 
2 −2
= −2  −2
+ 1 2
/6
2 −1
1
−2 −2
2
−1
 1
−1

−1
 1
−1
= −2 



−1
1 −1
−1  =  0
1
−1
1
1


3
0
3
0
0
0 
− 1
/6
3
0
3

−1
−2
−2   0
1
0
2
−2
−2
−2
2
2
−1
−2
3

0
0
3
−1
0   −2
0
2
−1


−2
1
2 
+ 2 2
/6
2
−1
0
2
−2



−1
2 −1
0
−2
2  =  −1 −1
1  0
2
1
1
1
0



4 −2
2
2
2 −2
1 −1 
2 −2
= −2  −2
+ 1 2
/6
2 −1
1
−2 −2
2
−1
 1
−1


−1
 −1
1
1
−1
−1

−1
1
1

= −2 
4
2
−2

−1
−2
−1   0
2
0
1
1
1

1
2
2
2
2
2
2
0
1
0

0
3
3
2
4
−2

0
2
0   −2
4
0

0

+ 4 0
/6
0
−1
−2
3

0
0
−3
−1
0  2
0
2
−1


2
1
2 
+ 2 1
/6
2
−2
3
2
−1
1
2
2


−1
−1
−1  =  1
1
0

3 −3
0
3
0 
= −2  −3
−
/6
0
0
0
−1
 1
−1
1
1
2

0
2
0   −2
2
0

0

+ 2 0
/6
0
1
1
1
1
1
−1

3
−2
1



1
−2
1
0
−2
1  =  −1 −1
1  0
1
1
1
1
0


2 −2
2 −2
2
1 −1 
2 −2
+ 1  −2
/6
−1
1
2 −2
2
0
1
0


0
−2
0  2
2
0

0

+ 2 0
/6
0
0
1
0

0
3
3
1
1
−2
−1
−2
3
0
3
3

0
2 
/6
2

2
2 
/6
4

1
2 
/6
3

0
3 
/6
3

3
2 
/6
1

−1
−2 
/6
1

1
2 
/6
3

0
3 
/6
3

0
2 
/6
2

−2
−2 
/6
4

1
2 
/6
3

0
3 
/6
3
76
Functions of matrices



1
−1
−1  =  0
−1
1

3
0 −3
0
0 
= −2  0
−
/6
−3
0
3
−1
 −1
1

−1
 −1
1
−1
1
−1

−1
2
2

= −2 

−1
 −1
1
= −2 

−1
 −1
−1
−1
1
1

= −2 

−1
 −1
−1
−1
1
−1

= −2 

−1
 −1
−1
−1
−1
1

= −2 

1
2
2
2
2
2
2

0
0
−3
−1
0  2
0
2
1


2
1
2 
+ 2  −2
/6
2
1



1
−2
1
0
−2
2  =  −1 −1
1  0
2
1
1
1
0


2 −2
2 −2
2
1 −1 
2 −2
+ 1  −2
/6
−1
1
2 −2
2
0
2
−2
−2
4
−2

0
−2
0  2
4
0

0

+ 4 0
/6
0
−1
−2
3




1
1 −1
1
−2
0
0
3
−1  =  1
1 −1   0 −1
0   −2
1
0
1
2
0
0
2
1




3
3
0
2 −2 −2
1
3
3
0 
2
2 
− 1  −2
+ 2  −1
/6
/6
0
0
0
−2
2
2
2
3
2
−1




−1
1 −1 −1
−2
0
0
3
1  =  0 −1
2   0 −1
0   −2
−1
1
1
1
0
0
2
−1




3
0
3
2
2 −2
1
0
0
0 
2 −2 
− 1 2
+ 2  −2
/6
/6
3
0
3
−2 −2
2
−1
0
−2
2



−1
2 −1
0
−2
−1  =  1
1 −1   0
1
1
1
1
0


4
2
2
2 −2 −2
2
1
1 
2
2
+ 1  −2
/6
2
1
1
−2
2
2

0
2
0   −2
2
0

0

+ 2 0
/6
0
1
2
−3




−1
1
1 −1
−2
0
0
3
1  =  1 −1
1   0 −1
0  2
1
0
1
2
0
0
2
−1




3
3
0
2 −2
2
1
3
3
0 
2 −2 
− 1  −2
+ 2  −1
/6
/6
0
0
0
2 −2
2
−2
3
−2
1
4
2
−2
−1
−1
−1


1
−2
−2   0
1
0
1
1
1
0
1
0

0
1
0

0
3
3
−1
1
−2
−2
4
2
0
3
−3
−1
1
2

3
2 
/6
1

1
−2 
/6
1

1
2 
/6
3

0
3 
/6
3

0
2 
/6
2

2
−2 
/6
4

3
2 
/6
1

−1
2 
/6
1

1
2 
/6
3

0
−3 
/6
3

0
2 
/6
2

−2
2 
/6
4
1.7. 3 × 3 symmetric matrices

77



−1
1 −2
0
−1
3  =  −1 −1
1  0
3
1
1
1
0



2 −2
2
4
2 −2
2 −2 
1 −1
= −1  −2
+ 2 2
/6
2 −2
2
−2 −1
1
1
 1
−1


1
3
3
1
1
3
1
3
1

= −2 

1
 −1
1
−1
3
3

= −1 

1
 −1
3
2
2
−2
−1
1
3

= −3 

3
0
−3
2
2
−2


−1
3
−1

1
3
1
3
1
1

3
= −2  −3
0

0
2
0   −2
6
0

0

+ 6 0
/6
0
−2
−1
3
0
3
3



3
−1
1
1
−2
1  =  0 −2
1  0
1
1
1
1
0


0 −3
1 −2
1
0
0 
4 −2
+ 2  −2
/6
0
3
1 −2
1
0
2
0


0
−3
0  1
5
2

2

+ 5 2
/6
2
0
−2
2



1
−1
2
0
−1
3  =  −1 −1
1  0
3
1
1
1
0


2 −2
4 −2
2
2 −2 
1 −1
+ 2  −2
/6
−2
2
2 −1
1
0
2
0


0
−2
0  2
6
0

0

+ 6 0
/6
0
−2
−1
3




3
−1 −1
1
−3
0
0
−2
3  =  −1
1
1  0
2
0   −3
3
1
0
2
0
0
6
1




2 −2
3 −3
0
1
2 −2 
3
0 
+ 2  −3
+ 6 1
/6
/6
−2
2
0
0
0
2
−2
3
1


3
−1
−1  =  0
1
1

3
0 −3
0
0 
= −2  0
+
/6
−3
0
3
1
 −1
3
0
2
0


1
−2
−1   0
1
0
1
2
1

2
1
2
1
2
4
2

0
−3
0  1
5
2

1
2
2 
+ 5  −2
/6
1
2



1
−1 −1
1
−2
1  =  1 −1
1  0
3
0
2
1
0


−3
0
1
1 −2
3
0 
1 −2
+ 2 1
/6
0
0
−2 −2
4
2
2
2
0
3
3
1
1
2
0
2
0

0
2
−2

0
−3
0   −1
5
2

2

+ 5 2
/6
2
3
−1
2
0
2
0

−2
2
−2
2
2
2

2
1 
/6
3

0
3 
/6
3

3
1 
/6
2

2
2 
/6
2

2
1 
/6
3

0
3 
/6
3

2
0 
/6
2

2
2 
/6
4

3
1 
/6
2

2
−2 
/6
2

0
2 
/6
2

2
2 
/6
2
78
Functions of matrices



−1
−1
−1  =  1
3
0

3 −3
0
3
0 
= −2  −3
+
/6
0
0
0
1
 3
−1
3
1
−1



−1
−2
−1   0
1
0
1
1
2

0
−3
0  1
5
−2

2
2
2 
+ 5 2
/6
4
−2
0
2
0

3
1
−2

0
2
0   −3
6
1

1

+ 6 2
/6
1
−2
0
2
0
2
0


0
−2
0  0
7
2

0
2
−3 
+ 7 2
/6
3
2
1
−3
2
0
3
0


0
2
0   −3
5
1

1

+ 5 2
/6
1
−2
0
2

0
−3
0  1
7
2

2

+ 7 2
/6
2
0
−2
2




2
−1 −1
1
−1
0
0
−2
2  =  −1
1
1  0
3
0   −3
3
1
0
2
0
0
5
1




2 −2
3 −3
0
1
2 −2 
3
0 
+ 3  −3
+ 5 1
/6
/6
−2
2
0
0
0
2
−2
3
1

2
1
1
2
1
1
2



−1
1 −1
1
−3
3  =  −1
0
2  0
1
1
1
1
0



2 −2
2
3
0 −3
2 −2 
0
0
= −3  −2
+ 2 0
/6
2 −2
2
−3
0
3
1
 3
−1


3
3
3
1
3
3
3
3
1

4
= −2  −2
−2



3
−2
0
1  =  1 −1
3
1
1


−2 −2
1
1 
+ 2
/6
1
1

1
−2
1  0
1
0
0
0
0
0
3
−3



−1
1 −1
1
−1
2  =  −1
0
2  0
2
1
1
1
0



2 −2
2
3
0 −3
2 −2 
0
0
= −1  −2
+ 3 0
/6
2 −2
2
−3
0
3
2
 2
−1


2
3
2
2
2
3
2
3
2

= −1 

2
 −1
2
3
0
−3
−1
2
2

= −1 
2
2
−2



3
−1
1
1
−1
2  =  0 −2
1  0
2
1
1
1
0


0 −3
1 −2
1
0
0 
4 −2
+ 1  −2
/6
0
3
1 −2
1
0
2
0

0
1
0

2
2
−2
2
4
2
2
2
2
2
4
2
2
2
2
1
1
2

0
2 
/6
2

−2
−2 
/6
2

2
3 
/6
1

1
2 
/6
1

1
3 
/6
2

2
2 
/6
2

2
3 
/6
1

1
2 
/6
1

3
1 
/6
2

2
2 
/6
2

2
0 
/6
2

2
2 
/6
4
1.7. 3 × 3 symmetric matrices


2
3
2
3
2
2

3
= −1  −3
0


2
3
3
3
3
2

4
= −1  −2
−2

79



2
−1 −1
1
−1
2  =  1 −1
1  0
3
0
2
1
0


−3
0
1
1 −2
3
0 
1 −2
+ 1 1
/6
0
0
−2 −2
4
0
1
0


0
−3
0   −1
7
2

2

+ 7 2
/6
2
3
−1
2


3
−2
0
2  =  1 −1
3
1
1


−2 −2
1
1 
+ 1
/6
1
1
0
1
0


0
−2
0  0
8
2

0
2
−3 
+ 8 2
/6
3
2
1
−3
2

0
2
0   −2
6
0

0

+ 6 0
/6
0
−1
−2
3




3
−1 −1
1
−3
0
0
−2
3  =  −1
1
1   0 −2
0   −3
3
1
0
2
0
0
6
1




2 −2
3 −3
0
1
2 −2 
3
0 
− 2  −3
+ 6 1
/6
/6
−2
2
0
0
0
2
−2
3
1

1
−1
1  0
1
0
0
0
0
0
3
−3



−1
2 −1
0
−2
3  =  −1 −1
1  0
3
1
1
1
0



4 −2
2
2
2 −2
1 −1 
2 −2
= −2  −2
+ 1 2
/6
2 −1
1
−2 −2
2
−1
 1
−1

−1
 1
3
1
3
3
1
−1
3

= −3 

2
2
−2



3
−1
1
1
−4
1  =  0 −1
2  0
−1
1
1
1
0



3
0 −3
2 −2
2
0
0 
2 −2
= −4  0
+ 1  −2
/6
−3
0
3
2 −2
2
−1
 1
3

−1
 2
2
1
3
1
2
−1
2

3
= −3  −3
0
0
1
0

2
2
2
2
2
2
0
3
3
1
1
2

0
−3
0  2
4
1

1

+ 4 2
/6
1
0
−2
2




2
−1 −1
1
−3
0
0
−3
2  =  1 −1
1   0 −1
0   −2
3
0
1
2
0
0
5
1




−3
0
2
2 −2
1
3
0 
2 −2 
− 1 2
+ 5 1
/6
/6
0
0
−2 −2
2
2
3
−2
1
0
1
0

2
4
2
1
1
2

0
2 
/6
2

2
2 
/6
2

1
3 
/6
2

2
2 
/6
2

1
2 
/6
3

0
3 
/6
3

2
0 
/6
2

2
2 
/6
4

3
2 
/6
1

1
2 
/6
1

0
2 
/6
2

2
2 
/6
4
80
Functions of matrices





2
−1
1
1
−3
0
0
−3
2  =  0 −1
2   0 −1
0  2
−1
1
1
1
0
0
5
1





3
0 −3
2 −2
2
1
0
0 
2 −2 
= −3  0
− 1  −2
+ 5 2
/6
/6
−3
0
3
2 −2
2
1
−1
 2
2

2
3
2
2
4
2

0
−2
0  2
6
0

0

+ 6 0
/6
0
−1
−2
3
0
1
0


0
−3
0  2
4
1

2
1
2 
+ 4  −2
/6
2
1
0
2
−2

0
−3
0   −2
4
1

1

+ 4 1
/6
2
3
−2
1




1
1 −1
1
−3
0
0
2
3  =  −1
0
2   0 −2
0   −3
−1
1
1
1
0
0
6
1





2 −2
2
3
0 −3
1
2 −2 
0
0 
= −3  −2
− 2 0
+ 6 2
/6
/6
2 −2
2
−3
0
3
1
−2
0
2
−1
 −1
1
−1
3
3

= −2 

4
2
−2



1
−2
1
0
−2
3  =  −1 −1
1  0
3
1
1
1
0


2 −2
2 −2
2
1 −1 
2 −2
+ 1  −2
/6
−1
1
2 −2
2
0
−2
2


3
−1
−1  =  0
−1
1

3
0 −3
0
0 
= −4  0
+
/6
−3
0
3
−1
 −1
3

−1
 3
1
−1
3
−1

3
−1
1

3
= −4  −3
0

−1
 3
1
3
3
3

1
−4
−2   0
1
0
1
1
1

1
2
2
2
2
2
2



1
−1 −1
1
−4
1  =  1 −1
1  0
3
0
1
2
0


−3
0
2
2 −2
3
0 
2 −2
+ 1 2
/6
0
0
−2 −2
2
0
1
0

0
1
0

0
3
3
−2
4
−2
1
1
2
2
4
2

3
2 
/6
1

1
2 
/6
1

1
2 
/6
3

0
3 
/6
3

3
2 
/6
1

1
−2 
/6
1

0
2 
/6
2

2
2 
/6
4

2
3 
/6
1

1
2 
/6
1