Page 212 - Advanced engineering mathematics

192 CHAPTER 7 Matrices and Linear Systems

then
AX = x 1 A 1 + x 2 A 2 + ··· + x k A k .
For example,
⎛ ⎞
x 1

6 −34 6x 1 − 3x 2 + 4x 3
⎝ x 2 ⎠ =
2 1 7 2x 1 + x 2 + 7x 3
x 3

6 −3 4
= x 1 + x 2 + x 3 .
2 1 7
7.1.2 Terminology and Special Matrices

We will define some terms and special matrices that are encountered frequently.

The n × m zero matrix O nm is the n × m matrix having every element equal to zero.

For example

000
O 23 = .
000
If A is n × m then
A + O nm = O nm + A = A.

The negative of a matrix A is just the scalar product (−1)A formed by multiplying each
matrix element by −1. We denote this matrix −A.If B has the same dimensions as A, then we
denote B + (−A) as B − A, as we do with numbers.

A square matrix is one having the same number of rows and columns. If A =[a ij ] is n × n,
the main diagonal of A consists of the matrix elements a 11 ,a 22 ,··· ,a nn . These are the
matrix elements along the diagonal from the upper left corner to the lower right corner.

The n × n identity matrix is the n × n matrix I n having each i, j element equal to zero if
i = j, and each i,i element equal to 1. For example,
1000
⎛ ⎞
0100
⎜ ⎟
I 4 = ⎜ ⎟ .
⎝ 0010 ⎠
0001
Thus I n has 1 down the main diagonal and zeros everywhere else.

THEOREM 7.2

If A is n × m, then
I n A = AI m = A.

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October 14, 2010 14:23 THM/NEIL Page-192 27410_07_ch07_p187-246