College Algebra, 2013a

586 Systems of Equations and Matrices
The identity matrix is an example of what is called a square matrix as it has the same number
of rows as columns. Note that to in order to verify that the identity matrix acts as a multiplicative
identity, some care must be taken depending on the order of the multiplication. For example, take
the matrix 2 × 3 matrix A from earlier
[
A =
2 0 −1
−10 3 5
In order for the product I k A to be defined, k = 2; similarly, for AI k to be defined, k =3. Weleave
it to the reader to show I 2 A = A and AI 3 = A. In other words,
]
[ 1 0
0 1
][
2 0 −1
−10 3 5
]
=
[
2 0 −1
−10 3 5
]
and
[
2 0 −1
−10 3 5
] ⎡ ⎣
1 0 0
0 1 0
0 0 1
⎤
⎦ =
[
2 0 −1
−10 3 5
]
While the proofs of the properties in Theorem 8.5 are computational in nature, the notation becomes
quite involved very quickly, so they are left to a course in Linear Algebra. The following example
provides some practice with matrix multiplication and its properties. As usual, some valuable
lessons are to be learned.
Example 8.3.2.
1. Find AB for A =
[ −23 −1 17
46 2 −34
2. Find C 2 − 5C +10I 2 for C =
[ 1 −2
3 4
⎡
]
and B = ⎣
−3 2
1 5
−4 3
3. Suppose M is a 4 × 4 matrix. Use Theorem 8.5 to expand (M − 2I 4 )(M +3I 4 ).
]
⎤
⎦
Solution.
1. We have AB =
[ −23 −1 17
46 2 −34
] ⎡ ⎣
−3 2
1 5
−4 3
⎤
⎦ =
[ 0 0
0 0
2. Just as x 2 means x times itself, C 2 denotes the matrix C times itself. We get
]