Page 71 - A Course in Linear Algebra with Applications

2.3: Elementary Matrices 55

-1/2 0 |
1 -2/3 I
-1 2 I

1 0 -1/3 | 2/3 1/3 0'
0 1 -2/3 J 1/3 2/3 0
0 0 4/3 | 1/3 2/3 1

1/3 | 2/3 1/3 0
2/3 I 1/3 2/3 0
1 j 1/4 1/2 3/4

| 3/4 1/2 1/4 \
I V 2 1 V 2 ,
I 1/4 1/2 3/4/

which is the reduced row echelon form. Therefore A is invert-
ible and
/3/4 1/2 1/4'
A' 1 = 1 / 2 1 1/2
\ l / 4 1/2 3/4

This answer can be verified by checking that A A" 1 = 1$ =
l
A~ A.
As this example illustrates, the procedure for finding the
inverse of a n x n matrix is an efficient one; in fact at most
n 2 row operations are required to complete it (see Exercise
2.3.10).

Exercises 2.3

1. Express each of the following matrices as a product of
elementary matrices and its reduced row echelon form: