Page 36 - A Course in Linear Algebra with Applications

20 Chapter One: Matrix Algebra

Theorem 1.2.3
(a) If A is an inveriible matrix, then A - 1 is invertible
1 1
and {A' )- =A.
(b) If A and B are invertible matrices of the same size,
1
1
then AB is invertible and (AB)~ l = B~ A~ .
Proof
1 X
(a) Certainly we have AA~ = I — A~ A, equations which
x
can be viewed as saying that A is an inverse of A~ . Therefore,
since A~ x cannot have more than one inverse, its inverse must
be A.
1
(b) To prove the assertions we have only to check that B~ A~ 1
1
l
is an inverse of AB. This is easily done: (AB)(B~ A~ ) =
1
1
A(BB~ )A~ , by two applications of the associative law;
the latter matrix equals AIA~ l — AA~ l — I. Similarity
1
1
(B~ A~ )(AB) = I. Since inverses are unique, (AB)" 1 =
l
B~ A-\
Partitioned matrices
A matrix is said to be partitioned if it is subdivided into
a rectangular array of submatrices by a series of horizontal or
vertical lines. For example, if A is the matrix (aij)^^, then

/ a n a i2 | ai3 \
021 0.22 I CI23

\ a 3 i a 32 | a 33 /

is a partitioning of A. Another example of a partitioned matrix
is the augmented matrix of the linear system whose matrix
form is AX — B ; here the partitioning is [-A|S].
There are occasions when it is helpful to think of a matrix
as being partitioned in some particular manner. A common
one is when an m x n matrix A is partitioned into its columns
A±, A2, • • •, A n,