College Algebra 9th txtbk

472 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
Z DEFINITION 2 Inverse of a Square Matrix
If A is a square matrix of order n and if there exists a matrix
such that
A 1 A AA 1 I
(read “A inverse”)
then A 1 is called the multiplicative inverse of A or, more simply, the inverse of A.
If no such matrix exists, then A is said to be a singular matrix.
A 1
ZZZ EXPLORE-DISCUSS 2
Let
A c 4 2
2 2 d
B c 1
⁄4
1
⁄2
1
⁄2
1
⁄2
d
C c
1
⁄2 1 ⁄2
1 ⁄2
1 d
(A) How are the entries in A and B related?
(B) Find AB. Is B the inverse of A?
(C) Find AC. Is C the inverse of A?
The multiplicative inverse of a nonzero real number a also can be written as 1/
a, but
this notation is never used for matrix inverses.
Let’s use Definition 2 to find A 1 , if it exists, for
We are looking for a matrix
such that
We can write
is the identity matrix I. Mul-
and try to find a, b, c, and d so that the product of A and
tiplying A and A 1 on the left side, we get
A
A c 2 3
1 2 d
A 1 c a
b
AA 1 A 1 A I
A 1
c
d d
c 2 3
1 2 dca c
b d d c 1 0
0 1 d
(2a 3b) (2c 3d )
c
(a 2b) (c 2d) d c 1 0
0 1 d
I
A 1
which is true only if
2a 3b 1
a 2b 0
2c 3d 0
c 2 3 1
c 2 3 0
`
`
1 2 1 d
c 1 2 0
c 1 2` 1
`
2 3 1 d 2 3 0 d
R 1 4 R 2 R 1 4 R 2
1 2 0 d c 2d 1
Use Gauss–Jordan
elimination to solve
each system.
2R 1 R 2 S R 2 2R 1 R 2 S R 2