Page 211 - Advanced engineering mathematics

7.1 Matrices 191

Matrix addition and multiplication can be done in MAPLE using the A+B and A.B com-
mands, which are in the linalg package of subroutines. Multiplication of A by a scalar c is
achieved by c * A.

7.1.1 Matrix Multiplication from Another Perspective
Let A be an n × k matrix and B a k × m matrix. We have defined AB to be the n × m matrix
whose i, j-element is the dot product of row i of A with column j of B.
It is sometimes useful to observe that column j of AB is the matrix product of A with column
j of B. We can therefore compute a matrix product AB by multiplying an n × k matrix A in turn
by each k × 1 column of B.
Specifically, if the columns of B are B 1 ,··· ,B m , then we can think of B as a matrix of these
columns:
⎛ ⎞
···
··· B m .
B = B 1 B 2 ⎠
⎝
···
Then
⎛ ⎞
···
AB = A B 1 B 2 ··· B m ⎠
⎝
···
⎛ ⎞
···
··· AB m .
= AB 1 AB 2 ⎠
⎝
···
As an example, let

2 −4 −3 6 7
A = and B = .
1 7 −5 1 2
Then

2 −4 −3 14
= ,
1 7 −5 −38

2 −4 6 8
= ,
1 7 1 13
and
⎛ ⎞
8

2 −4 7
= ⎝ 6 ⎠ .
1 7 2
21
These are the columns of AB:

2 −4 −3 6 7 14 8 6
= .
1 7 −5 1 2 −38 13 21
We also will sometimes find it useful to think of a product AX, when X is a k × 1 column
matrix, as a linear combination of the columns A 1 ,··· ,A k of A. In particular, if
⎛ ⎞
x 1
x 2
⎜ ⎟
X = ⎜ . ⎟,
⎜ ⎟
.
⎝ . ⎠
x k

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October 14, 2010 14:23 THM/NEIL Page-191 27410_07_ch07_p187-246