Page 201 - Discrete Mathematics and Its Applications

180 2 / Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

both defined, they will not be the same size unless m = n = r = s. Hence, if both AB and BA
are defined and are the same size, then both A and B must be square and of the same size.
Furthermore, even with A and B both n × n matrices, AB and BA are not necessarily equal, as
Example 4 demonstrates.

EXAMPLE 4 Let

1 1 2 1
A = and B = .
2 1 1 1

Does AB = BA?

Solution: We find that

3 2 4 3
AB = and BA = .
5 3 3 2
Hence, AB = BA. ▲

Transposes and Powers of Matrices

We now introduce an important matrix with entries that are zeros and ones.

DEFINITION 5 The identity matrix of order n is the n × n matrix I n =[δ ij ], where δ ij = 1if i = j and
δ ij = 0if i = j. Hence

⎡ ⎤
1 0 ... 0
⎢0 1 ... 0⎥
⎢ ⎥
⎢· · ·⎥
⎥ .
I n = ⎢
⎢ · · · ⎥
⎣ ⎦
· · ·
0 0 ... 1

Multiplying a matrix by an appropriately sized identity matrix does not change this matrix. In
other words, when A is an m × n matrix, we have

AI n = I m A = A.

Powers of square matrices can be defined. When A is an n × n matrix, we have

0
r
A = I n , A = AAA ··· A .

r times
The operation of interchanging the rows and columns of a square matrix arises in many
contexts.