Page 210 - Advanced engineering mathematics

190 CHAPTER 7 Matrices and Linear Systems

THEOREM 7.1

Let A, B and C be matrices. Then, whenever the indicated operations are defined:
1. A + B = B + A (matrix addition is commutative).
2. A(B + C) = AB + AC.
3. (A + B)C = AC + AC.
4. (AB)C = A(BC).
5. cAB = (cA)B = A(cB) for any scalar c.

Proof Proofs of these conclusions are straightforward. To illustrate, we will prove opera-
tion (3):
i, j element of A(B + C) = (row i of A) · (column j of B + C)

= (row i of A) · (column j of B + column j of C)
= (row i of A) · (column j of B) + ((row i of A) · (column j of C)
= (i, j element of AB) + (i, j element of AC)
= i, j element of AB + AC.

We have already noted that in some ways matrix multiplication does not behave like
multiplication of real numbers. The following examples illustrate other differences.

EXAMPLE 7.3
Even when AB and BA are defined and have the same dimensions, it is possible that AB = BA:

1 0 −2 6 −2 0
=
2 −4 1 3 8 0
but

−20 1 0 −14 24
= .
8 0 2 −4 −5 12
EXAMPLE 7.4
There is in general no cancelation in products: if AB = AC, it does not follow that A = C.To
illustrate,

11 4 2 11 2 7 7 18
= = ,
33 316 33 511 21 54
even though
4 2 2 7

= .
316 5 11
EXAMPLE 7.5
The product of two nonzero matrices may be a zero matrix:

12 6 4 00
= .
00 −3 −2 00

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October 14, 2010 14:23 THM/NEIL Page-190 27410_07_ch07_p187-246