Page 209 - Advanced engineering mathematics

7.1 Matrices 189

This clarifies why the number of columns of A must equal the number of rows of B for
the product AB to be defined. We can only take the dot product of two vectors of the same
dimension.

EXAMPLE 7.1
Let

13 1 1 3
A = and B = .
25 2 1 4
Here A is 2 × 2 and B is 2 × 3, so we can compute AB, which is 2 × 3 (number of rows of A,
number of columns of B). In terms of dot products of rows with columns,

13 1 1 3
AB =
25 2 1 4

< 1,3 > · < 1,2 > < 1,3 > · < 1,1 > < 1,3 > · < 3,4 >
=
< 2,5 > · < 1,2 > < 2,5 > · < 1,1 > < 2,5 > · < 3,4 >

7 4 15
= .
12 7 26
In this example, BA is not defined because the number of columns of B does not equal the number
of rows of A.

EXAMPLE 7.2
Let
⎛ ⎞
−18

1121 ⎜ 2 1 ⎟
A = and B = ⎜ ⎟ .
4162 ⎝ 1 1 ⎠
12 6
Because A is 2 × 4 and B is 4 × 2, then AB is defined and is 2 × 2:

< 1,1,2,1 > · < −1,2,1,12 > < 1,1,2,1 > · < 8,1,1,6 >
AB =
< 4,1,6,2 > · < −1,2,1,12 > < 4,1,6,2 > · < 8,1,1,6 >

15 17
= .
28 51
In this example, BA is also defined and is a 4 × 4 matrix:
⎛ ⎞ ⎛ ⎞
−1 8 31 7 46 15

2 1 1121 6 3 10 4
⎜ ⎟ ⎜ ⎟
BA = ⎜ ⎟ = ⎜ ⎟ .
⎝ 1 1 ⎠ 4162 ⎝ 5 2 8 3 ⎠
12 6 36 18 60 24
Even when both AB and BA are defined, these matrices may not be equal, and may not even
have the same dimensions. Matrix multiplication is noncommutative.

We will list some properties of these matrix operations.

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October 14, 2010 14:23 THM/NEIL Page-189 27410_07_ch07_p187-246