Page 127 - Matrix Analysis & Applied Linear Algebra

3.7 Matrix Inversion 121

In general the product of two rank-r matrices does not necessarily have to
produce another matrix of rank r. For example,

12 2 4
A = and B =
24 −1 −2
each has rank 1, but the product AB = 0 has rank 0. However, we saw in
(3.7.14) that the product of two invertible matrices is again invertible. That is, if
rank (A n×n )= n and rank (B n×n )= n, then rank (AB)= n. This generalizes
to any number of matrices.

Products of Nonsingular Matrices Are Nonsingular

If A 1 , A 2 ,..., A k are each n × n nonsingular matrices, then the prod-
uct A 1 A 2 ··· A k is also nonsingular, and its inverse is given by the
reverse order law. That is,
−1 −1 −1 −1
(A 1 A 2 ··· A k ) = A ··· A A .
k 2 1

Proof. Apply (3.7.14) and (3.7.15) inductively. For example, when k =3 you
can write
−1 −1 −1 −1 −1 −1
(A 1 {A 2 A 3 }) = {A 2 A 3 } A = A A A .
1 3 2 1
Exercises for section 3.7

3.7.1. When possible, find the inverse of each of the following matrices. Check
your answer by using matrix multiplication.
 
4 −85
12 12
(a) (b) (c)  4 −74 
13 24
3 −42
 
  1111
123
(d)  456  (e)  1233 
 1222 
789
1234
3.7.2. Find the matrix X such that X = AX + B, where
   
0 −1 0 12
A =  0 0 −1  and B =  21  .
0 0 0 33