Page 103 - Matrix Analysis & Applied Linear Algebra

3.5 Matrix Multiplication 97

For example,
 
13 −32
21 −4 83 −74
A = , B =  25 −18  =⇒ AB = .
−30 5 −81 94
−12 02
Notice that in spite of the fact that the product AB exists, the product BA
is not defined—matrix B is 3 × 4 and A is 2 × 3, and the inside dimensions
don’t match in this order. Even when the products AB and BA each exist
and have the same shape, they need not be equal. For example,

1 −1 11 00 2 −2
A= , B= =⇒ AB= , BA= . (3.5.1)
1 −1 11 00 2 −2
This disturbing feature is a primary difference between scalar and matrix algebra.

Matrix Multiplication Is Not Commutative
Matrix multiplication is a noncommutative operation—i.e., it is possible
for AB
= BA, even when both products exist and have the same shape.

There are other major differences between multiplication of matrices and
multiplication of scalars. For scalars,
αβ = 0 implies α =0 or β =0. (3.5.2)

However, the analogous statement for matrices does not hold—the matrices given
in (3.5.1) show that it is possible for AB = 0 with A
= 0 and B
= 0. Related
to this issue is a rule sometimes known as the cancellation law. For scalars,
this law says that
αβ = αγ and α
= 0 implies β = γ. (3.5.3)

This is true because we invoke (3.5.2) to deduce that α(β − γ) = 0 implies
β − γ =0. Since (3.5.2) does not hold for matrices, we cannot expect (3.5.3) to
hold for matrices.
Example 3.5.1

The cancellation law (3.5.3) fails for matrix multiplication. If

11 22 31
A = , B = , and C = ,
11 22 13
then

44
AB = = AC but B
= C
44
in spite of the fact that A
= 0.