2. Matrix Algebra
2.1 Matrix Operations
j-th column
 
A  ai , j
 a11
 

  ai1

 
am1

 a1 j
 
 aij

 amj
 a1n 

 
 ain 

  
 amn 
i-th row
Diagonal entries
Diagonal matrix : a square matrix
whose nondiagonal entries are zero.
Recall: Two matrices are equal

the matrices are the same size and
their corresponding entries are equal.
Theorem 1
Let A, B, and C be matrices of the same size, and let r and
s be scalars.
A B  B  A
( A  B)  C  A  ( B  C )
A0  A
r ( A  B )  rA  rB
( r  s ) A  rA  sA
r ( sA)  ( rs ) A
Example:
0 2
1
A :  3 1 0
 4  2 1
 2  1 1
B :  2 1 0
 3 0 2
A B 
A B 
2B 
1
 A  3B 
2
A2 
Example:
 1


A :=  4


 -7
-2
-5
8
-3 


6


9
 1


 -7
C := 
 4


 0
-3
9
-6
0
5


-2 


8


0
AC 
AC 
2C 
1
 A  3C 
2
Matrix Multiplication
REVIEW
x1 
 
x 2 

Ax  a1 a 2 L a n 
 x1a1  x 2a 2 L  x n a n
 M
 
x n 
Recall:
1
1 3 2  1 1  3 1 

          
Ax  2
0 2 1 2210  0 2  4 

3 2 2
 
0 
 
3
 
2
 
2 
 
8 


Matrix Multiplication
If A is an m  n matrix and B is an n  p matrix
with columns b1,b2 ,L ,b p , then the product
AB is the m  p matrix whose columns are A b1,K ,Ab p

 
AB  A b1 b2 L b p  A b1 A b2 L A b p
Example: Let
Find AB 
1 2 
A  0  1
 4 5 
2 3 
B

1  2 

Row-Column Rule for Computing AB: If the product
AB is defined, then the entry in row i and column j of AB
is the sum of the products of corresponding entries
from row i of A and column j of B.
( AB) ij  ai1b1 j  ai 2b2 j    aipb pj    ainbnj
Example:
1 2 
 _______ _______ 
0  1  2 3   _______ _______ 

 1  2  

 4  2  5 1 _______ 
 4 5  


3x2
2x2
3x2
Properties of Matrix Multiplication
A( BC )  ( AB)C
A( B  C )  AB  AC
( B  C ) A  BA  CA
r ( AB)  (rA) B  A(rB ) for any scalar r
I m A  A  AI n for m  n matrix A
In general the followings are NOT true.
AB  BA
If AB  AC then B  C
If AB  0 then A  0 or B  0
Defn: Given an m×n matrix A, the transpose of A is
the n×m matrix, denoted by AT, whose columns are
formed from the corresponding rows of A.
A  (aij )  AT  (a ji )
 3 1 0
Example: Let A : 


1
2
4


What is AT ?
Rules related to transpose:
(A )  A
T T
( A  B)  A  B
T
T
T
(rA)T  rAT for any scalar r
( AB)T  B T AT