Page 443 - Schaum's Outline of Theory and Problems of Signals and Systems

REVIEW OF MATRIX THEORY [APP. A

d. Multiplication:
Let A = [a,,],.,, B = [bijInxp, and C = [cijImxp.
n
C=AB =Cij= aikbkj
k=l
The matrix product AB is defined only when the number of columns of A is equal to the number of
rows of B. In this case A and B are said to be conformable.
EXAMPLE A.2 Let

Then

0(1)+(-1)3 0(2)+(-I)(-1)
1(1) + 2(3)
2(1) + ( -3)3 2(2) + ( -3)( - 1)

but BA is not defined.

Furthermore, even if both AB and BA are defined, in general
AB # BA

EXAMPLE A.3 Let
-;] B -;I
A
B=[; [ -:I=[-: A]
Then

2 0 -:I=[-: -:]-
.A=[:

An example of the case where AB = BA follows.

EXAMPLE A.4 Let
1 0 2 0
0 31 ~=[o 41

Then AB=BA=[~

Notes:

1. AO=OA=O
2. A1 = LA = A
3. (A + B)C = AC + BC
4. A(B+C)=AB+AC
5. (AB)C = A(BC) = ABC
6. a(AB) = (aA)B = A(aB)