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**Unformatted text preview: **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
2
−10 A= 0 −1
3
5 In order for the product Ik A to be deﬁned, k = 2; similarly, for AIk to be deﬁned, k = 3. We leave
it to the reader to show I2 A = A and AI3 = A. In other words,
10
01 2
−10 0 −1
3
5 = 2
−10 0 −1
3
5 and 100
0 1 0 =
001 2
−10 0 −1
3
5 2
−10 0 −1
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 = −3 2
and B = 1 5 −4 3 −23 −1
17
46
2 −34 2. Find C 2 − 5C + 10I2 for C = 1 −2
3
4 3. Suppose M is a 4 × 4 matrix. Use Theorem 8.5 to expand (M − 2I4 ) (M + 3I4 ).
Solution. 1. We have AB = −23 −1
17
46
2 −34 −3 2 1 5 =
−4 3 00
00 2. Jus...

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