A First Course in Linear Algebra, 2017a

72 Matrices
Theorem 2.34: Uniqueness of Inverse
Suppose A is an n × n matrix such that an inverse A −1 exists. Then there is only one such inverse
matrix. That is, given any matrix B such that AB = BA = I, B = A −1 .
Proof. In this proof, it is assumed that I is the n × n identity matrix. Let A,B be n × n matrices such that
A −1 exists and AB = BA = I. We want to show that A −1 = B. Now using properties we have seen, we get:
A −1 = A −1 I = A −1 (AB)= ( A −1 A ) B = IB = B
Hence, A −1 = B which tells us that the inverse is unique.
♠
The next example demonstrates how to check the inverse of a matrix.
Example 2.35: Verifying the Inverse of a Matrix
[ ] [ ]
1 1
2 −1
Let A = . Show
is the inverse of A.
1 2
−1 1
Solution. To check this, multiply
[
1 1
1 2
][
2 −1
−1 1
]
=
[
1 0
0 1
]
= I
and [
2 −1
−1 1
][ 1 1
1 2
showing that this matrix is indeed the inverse of A.
]
=
[ 1 0
0 1
]
= I
♠
Unlike ordinary multiplication of numbers, it can happen that A ≠ 0butA mayfailtohaveaninverse.
This is illustrated in the following example.
Example 2.36: A Nonzero Matrix With No Inverse
[ ]
1 1
Let A = . Show that A does not have an inverse.
1 1
Solution. One might think A would have an inverse because it does not equal zero. However, note that
[ ][ ] [ ]
1 1 −1 0
=
1 1 1 0
If A −1 existed, we would have the following
[ ]
0
0
])
= A −1 ([
0
0