Page 247 - Advanced engineering mathematics

7.7 Matrix Inverses 227

It is easy to find matrices that have no inverse. For example, let

10
A = .
20
Suppose

a b
B = .
c d
is an inverse of A. Then

10 a b a b 10
AB = = = ,
20 c d 2a 2b 01
implying that
a = 1,b = 0,2a = 0 and b = 1

and this is impossible. On the other hand, some matrices do have inverses. For example,

21 4/7 −1/7 4/7 −1/7 21 10
= = .
14 −1/7 2/7 −1/7 2/7 14 01

A matrix that has an inverse is called nonsingular. A matrix with no inverse is singular.

A matrix can have only one inverse. For suppose that B and C are inverses of A. Then
B = BI n = B(AC) = (BA)C = I n C = C.
−1
In view of this, we will denote the inverse of A as A . Here are additional facts about nonsingular
matrices and matrix inverses.

THEOREM 7.15

Let A be an n × n matrix. Then,
1. I n is nonsingular and is its own inverse.
2. If A and B are nonsingular n × n matrices, then so is AB. Further,
−1
−1
−1
(AB) = B A .
The inverse of a product is the product of the inverses in the reverse order. This extends to a
product of any finite number of matrices.
−1
3. If A is nonsingular, so is A , and
−1 −1
(A ) = A.
The inverse of the inverse is the matrix itself.
4. If A is nonsingular, so is its transpose A , and
t
t −1
−1 t
(A ) = (A ) .
The inverse of a transpose is the transpose of the inverse.
5. A is nonsingular if and only if A R = I n .
6. A is nonsingular if and only if rank(A) = n.

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October 14, 2010 14:23 THM/NEIL Page-227 27410_07_ch07_p187-246