Definition of 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 (or element):
aij
number of rows: m
number of columns: n
size: m×n
Square matrix: m = n
2.2
Equal matrices: two matrices are equal if they have the same
size (m × n) and entries corresponding to the same position
are equal
For A  [aij ]m n and B  [bij ]mn ,
A  B if and only if aij  bij for 1  i  m, 1  j  n
Ex 1: Equality of matrices
1 2
A

3
4


a b 
B

c
d


If A  B, then a  1, b  2, c  3, and d  4
2.3
Matrix
addition:
If A  [aij ]m n , B  [bij ]m n ,
then A  B  [aij ]mn  [bij ]mn  [aij  bij ]mn  [cij ]mn  C
Ex 2: Matrix addition
 1 2  1 3  1  1 2  3  0 5
 0 1   1 2   0  1 1  2   1 3

 
 
 

 1  1
  3   3 
   
 2  2
 1  1  0 
  3  3   0 

  
 2  2 0
2.4
Scalar multiplication:
If A  [aij ]m n and c is a constant scalar,
then cA  [caij ]m n
Matrix subtraction:
A  B  A  (1) B
Ex 3: Scalar multiplication and matrix subtraction
 1 2 4
A   3 0  1


 2 1 2
0 0
 2
B   1  4 3


3 2
 1
Find (a) 3A, (b) –B, (c) 3A – B
2.5
Matrix multiplication:
If A  [aij ]m n and B  [bij ]n p ,
then AB  [aij ]m n [bij ]n p  [cij ]m p  C ,
should be equal
size of C=AB
n
where
cij   aik bkj  ai1b1 j  ai 2b2 j    ainbnj
k 1
 a11 a12  a1n  b  b
 b1n  
 11

1j
 


 
 


b

b

b
21
2
j
2
n


 ai1 ai 2  ain  



  ci1 ci 2  cij  cin 
 


 


 bn1  bnj  bnn  


an1 an 2  ann  
※ The entry cij is obtained by calculating the sum of the entry-byentry product between the ith row of A and the jth column of B
2.6
Ex 4: Find AB
Sol:
 1 3 
A   4 2 
 5 0  3 2
 3 2 
B


4
1

 2 2
 (1)(3)  (3)(4) (1)(2)  (3)(1) 
AB  (4)(3)  (2)(4) (4)(2)  ( 2)(1) 
 (5)(3)  (0)(4)
(5)(2)  (0)(1)  3 2
 9 1 
  4 6 
 15 10  3 2
Note: (1) BA is not multipliable
(2) Even BA is multipliable, AB≠BA
2.7
Matrix form of a system of linear equations in n variables:
 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
A xb
m  n n 1
m 1
2.8
Properties of Matrix Operations
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
Zero matrix :
Identity matrix of order n :
0mn
0 0
0 0



0 0
0
0 


0  mn
1 0
0 1
In  


0 0
0
0 


1  n n
2.9
Transpose of a matrix :
 a11
a
If A   21


 am1
a12
a22
am 2
 a11
a
then AT   12


 a1n
a21
a22
a2 n
a1n 
a2 n 
 M m n ,


amn 
am1 
am 2 
 M n m


amn 
※ The transpose operation is to move the entry aij (original at the
position (i, j)) to the position (j, i)
※ Note that after performing the transpose operation, AT is with the
size n×m
2.10
Ex 8: Find the transpose of the following matrix
(a)
Sol:
 2
A 
8 
(a)
(b)
(c)
(b)
 1 2 3
A   4 5 6


7 8 9
(c)
1
0
A  2 4


 1  1
 2
A 
 AT  2 8
8 
 1 2 3
1 4 7 
A  4 5 6  AT  2 5 8 




7 8 9
3 6 9 
1
0
A  2 4


 1  1
0 2 1 
A 

1
4

1


T
2.11
Symmetric matrix :a
A square matrix A is symmetric if A = AT
Skew-symmetric matrix :
A square matrix A is skew-symmetric if AT = –A
Ex:
Sol:
 1 2 3
If A  a 4 5


b c 6
is symmetric, find a, b, c?
 1 2 3
1 a b 
A  a 4 5 AT  2 4 c 




b c 6
3 5 6
A  AT
 a  2, b  3, c  5
2.12
Properties of transposes:
(1) ( AT )T  A
(2) ( A  B)T  AT  BT
(3) (cA)T  c( AT )
(4) ( AB)T  BT AT
※ Properties (2) and (4) can be generalized to the sum or
product of multiple matrices. For example, (A+B+C)T =
AT+BT+CT and (ABC)T = CTBTAT
※ Since a real number also can be viewed as a 1 × 1 matrix, the
transpose of a real number is itself,
, aT = a. In
a  R that is, for
other words, transpose operation has actually no function on
real numbers
2.13