CHAPTER 2
MATRICES
2.1 Operations with Matrices

Matrix
 a11 a12 a13  a1n 
a

a
a

a
22
23
2n 
 21
A  [aij ]   a31 a32 a33  a3n 
 M mn





 
am1 am 2 am 3  amn 
mn
(i, j)-th entry:
row: m
aij
column: n
size: m×n
2-1

i-th row vector
ri  ai1

ai 2  ain 
j-th column vector
 c1 j 
c 
2j
cj   
  
c 
 mj 

row matrix
column matrix
Square matrix: m = n
2-2
2-3
2-4
2-5
2-6

Matrix form of a system of linear equations:
 a11 x1  a12 x2    a1n xn  b1
 a x  a x  a x  b
 21 1 22 2
2n n
2



am1 x1  am 2 x2    amn xn  bm
m linear equations

 a11
a
 21
 

am1
a12
a22

am 2
 a1n   x1   b1 
 a2 n   x2   b2 


      
   
 amn   xn  bm 
=
=
=
A
x
b
Single matrix equation
Ax b
m  n n 1
m 1
2-7

Partitioned matrices:
submatrix
 a11
A  a21

a31
a12
a13
a22
a32
a23
a33
 a11
A  a21

a31
a14 
 A11

a24  
  A21
a34 
a12
a13
a22
a32
a23
a33
 a11
A12 
a
A

 21
A22 
a31
a14 
a24   c1 c2

a34 
c3
a12
a13
a22
a32
a23
a33
a14   r1 
a24   r2 
  
a34   r3 
c4 
2-8

a linear combination of the column vectors of matrix A:
 a11
a
A   21
 

am1
a12
a22

am 2
 x1 
x 
x   2

 
 xn 
 a1n 
 a2 n 
 c1 c2  cn 

 

 amn 
 a11 x1  a12 x2    a1n xn 
 a11 
 a21 
 a1n 
 a x  a x  a x 
a 
a 
a 
2n n 
 Ax   21 1 22 2
 x1  21   x2  22     xn  2 n 


  
  
  



 
 
 
a
x

a
x



a
x
a
a
mn n  m1
 m1 1 m 2 2
 m1 
 m2 
amn 
=
c2
=
=
 x1c1  x2c2   xncn
c1
cn
linear combination of
column vectors of A
2-9
2.2 Properties of Matrix Operations

Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication

Zero matrix: 0mn

Identity matrix of order n: I n
1
0



0
0
1

0




0
0


1
2-10
2-11
2-12
2-13

Real number:
ab = ba

(Commutative law for multiplication)
Matrix:
AB  BA
mn n p
Three situations:
(1) If m  p , then AB is defined，BA is undefined.
(2) If m  p, m  n, then AB  M mm，BA M nn (Sizes are not the same)
(3) If m  p  n, then AB  M mm，BA M mm
(Sizes are the same, but matrices are not equal)
2-14

Real number:
ac  bc , c  0
(Cancellation law)
 ab

Matrix:
AC  BC
C0
(1) If C is invertible, then A = B
(2) If C is not invertible, then A  B (Cancellation is not valid)
2-15
2-16

Transpose of a matrix:
 a11
a
If A   21
 
a
 m1
a12
a22

am 2
 a11
a
Then AT   12
 
a
 1n
 a1n 
 a2 n 
  M mn
 

 amn 
a21  am1 
a22  am 2 
  M nm


 
a2 n  amn 
2-17
2-18
2-19

Symmetric matrix:
A square matrix A is symmetric if A = AT


Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A
T
Note: AA is symmetric
Proof:
( AA )  ( A ) A  AA
T T
T T
T
T
 AAT is symmetric
2-20
2.3 The Inverse of a Matrix
Notes: AA 1  A1 A  I
2-21
A

-Jordan Elimination
| I  Gauss

 I | A1

If A can’t be row reduced to I, then A is singular.
2-22

Power of a square matrix:
(1) A0  I
(2) Ak  
AA

A



(k  0)
k factors
(3) Ar  As  Ar s
d1
0
(4) D  

0

r, s : integers
0
d1k

0
0
  Dk  



 d n 
0
0 
d2 
 
0
( Ar ) s  Ars
0
d 2k

0
0

0

k
 dn 


2-23
2-24

Note:
 A1 A2 A3  An 1  An1  A31 A21 A11
2-25

Note: If C is not invertible, then cancellation is not valid.
2-26
2-27
2.4 Elementary Matrices

Note:
Only do a single elementary row operation.
2-28
2-29
2-30
2-31

Note:
If A is invertible
Then Ek  E3 E2 E1 A  I
A1  Ek  E3 E2 E1
A  E11E21E31 Ek1
Ek  E3 E2 E1[ A I ]  [ I A1 ]
2-32
2-33
2-34

Note:
If a square matrix A can be row reduced to an upper triangular
matrix U using only the row operation of adding a multiple of
one row to another, then it is easy to find an LU-factorization of A.
Ek  E2 E1 A  U
A  E11 E21  Ek1U
A  LU
2-35

Solving Ax=b with an LU-factorization of A
Ax  b

If A  LU， then LUx  b
Let y  Ux， then Ly  b
Two steps:
(1) Write y = Ux and solve Ly = b for y
(2) Solve Ux = y for x
2-36