Page 113 - Matrix Analysis & Applied Linear Algebra

3.6 Properties of Matrix Multiplication 107

Proof. Notice that I ∗j has a 1 in the j th position and 0’s elsewhere. Recall
from Exercise 3.5.4 that such columns were called unit columns, and they
have the property that for any conformable matrix A,

AI ∗j = A ∗j .

Using this together with the fact that [AI] ∗j = AI ∗j produces

AI =( AI ∗1 AI ∗2 ··· AI ∗n )=( A ∗1 A ∗2 ··· A ∗n )= A.
A similar argument holds when I appears on the left-hand side of A.
Analogous to scalar algebra, we define the 0 th power of a square matrix to
be the identity matrix of corresponding size. That is, if A is n × n, then
0
A = I n .
Positive powers of A are also defined in the natural way. That is,

n
A = AA···A .

n times
The associative law guarantees that it makes no difference how matrices are
2
2
grouped for powering. For example, AA is the same as A A, so that
2
2
3
A = AAA = AA = A A.
Also, the usual laws of exponents hold. For nonnegative integers r and s,
r s
rs
s
r
A A = A r+s and (A ) = A .
We are not yet in a position to define negative or fractional powers, and due to
the lack of conformability, powers of nonsquare matrices are never defined.
Example 3.6.2
2
A Pitfall. For two n × n matrices, what is (A + B) ? Be careful! Because
matrix multiplication is not commutative, the familiar formula from scalar alge-
bra is not valid for matrices. The distributive properties must be used to write

2
(A + B) =(A + B)(A + B)=(A + B) A +(A + B) B

2
2
= A + BA + AB + B ,
2
2
and this is as far as you can go. The familiar form A +2AB+B is obtained only
k
in those rare cases where AB = BA. To evaluate (A + B) , the distributive
rules must be applied repeatedly, and the results are a bit more complicated—try
it for k =3.