Page 250 - Advanced engineering mathematics

230 CHAPTER 7 Matrices and Linear Systems

then
.
⎛ ⎞
. . 2 3 . . 1 0
[A.I 2 ]= ⎝ ⎠ .
.
−19 . . 0 1
.
.
Reduce A, carrying out the row operations across the entire matrix [A.I n ]. A is nonsingular
−1
exactly when A R =I n turns up in the first n columns. In this event the second n columns form A .
EXAMPLE 7.28
Let

5 −1
A = .
6 8
Form
.
⎛ ⎞
. 5 −1 . . 1 0
.
[A.I 2 ]= ⎝ ⎠ .
.
6 8 . . 0 1
Reduce A, carrying out each row operation on the entire row of the augmented matrix. First
multiply row one by 1/5:
.
⎛ ⎞
1 −1/5 . . 1/5 0
.
⎝ ⎠ .
6 8 . . 0 1
Add −6 times row one to row two:
.
⎛ ⎞
1 −1/5 . . 1/5 0
⎝ ⎠ .
.
0 46/5 . . −6/5 1
Multiply row two by 5/46:
.
⎛ ⎞
1 −1/5 . . 1/5 0
.
⎝ ⎠ .
6 1 . . −6/46 5/46
Add 1/5 times row two to row one:
.
⎛ ⎞
10 . . 8/46 1/46
⎝ ⎠ .
.
01 . . −6/46 5/46
This is in reduced form. The first two columns are A R . Since A R = I 2 , A is nonsingular. Further,
we can read A −1 from the last two columns:

8/46 1/46
−1 .
A =
−6/46 5/46
EXAMPLE 7.29
Let

−3 21
A = .
4 −28
Form
.
⎛ ⎞
. −3 21 . . 10
.
[A.I 2 ]= ⎝ ⎠ .
.
4 −28 . . 01

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October 14, 2010 14:23 THM/NEIL Page-230 27410_07_ch07_p187-246