Page 35 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 19

is a candidate. To be sure that B is really an inverse of A, we
need to verify that BA is also equal to I2', this is in fact true,
as the reader should check.
At this point the natural question is: how can we tell if
a square matrix is invertible, and if it is, how can we find an
inverse? From the examples we have seen enough to realise
that the question is intimately connected with the problem of
solving systems of linear systems, so it is not surprising that
we must defer the answer until Chapter Two.
We now present some important facts about inverses of
matrices.
Theorem 1.2.2

A square matrix has at most one inverse.
Proof
Suppose that a square matrix A has two inverses B\ and B<i-
Then
AB X = AB 2 = 1 = B XA = B 2A.

The idea of the proof is to consider the product (BiA)B2\
since B\A = I, this equals IB 2 = B 2. On the other hand,
by the associative law it also equals Bi(AB2), which equals
BJ = B x. Therefore B x = B 2.

From now on we shall write

1
A-

for the unique inverse of an invertible matrix A.