There is a special matrix called I and referred to as the identity matrix. It is always a
square matrix, meaning the number of rows equals the number of columns and it has the
property that there are ones down the main diagonal and zeroes elsewhere. Here are some
identity matrices of various sizes.

The first is the 1 × 1 identity matrix, the second is the 2 × 2 identity matrix, the third is
the 3 × 3 identity matrix, and the fourth is the 4 × 4 identity matrix. By extension, you
can likely see what the n×n identity matrix would be. It is so important that there is a
special symbol to denote the ij^{th} entry of the identity matrix I_{ij} = δ_{ij} where δ_{ij} is the
Kronecker symbol defined by

{
δ = 1 if i = j
ij 0 if i ⁄= j

It is called the identity matrix because it is a multiplicative identity in the following
sense.

Lemma 18.6.1Suppose A is an m × n matrix and I_{n}is the n × n identitymatrix. Then AI_{n} = A. If I_{m}is the m × m identity matrix, it also follows thatI_{m}A = A.

Proof:

∑
(AIn)ij = Aikδkj = Aij
k

and so AI_{n} = A. The other case is left as an exercise for you. ■

Definition 18.6.2An n × n matrix A has an inverse, A^{−1}if and onlyif AA^{−1} = A^{−1}A = I. Such a matrix is calledinvertible.

It is very important to observe that the inverse of a matrix, if it exists, is unique.
Another way to think of this is that if it acts like the inverse, then it is the
inverse.

Theorem 18.6.3Suppose A^{−1}exists and AB = BA = I. Then B = A^{−1}.

Proof:

−1 −1 −1 ( −1 )
A = A I = A (AB ) = A A B = IB = B. ■

Unlike ordinary multiplication of numbers, it can happen that A≠0 but A may fail to
have an inverse. This is illustrated in the following example.

Example 18.6.4Let A =

( 1 1 )
1 1

. Does A have an inverse?

One might think A would have an inverse because it does not equal zero.
However,

( ) ( ) ( )
1 1 − 1 = 0
1 1 1 0

and if A^{−1} existed, this could not happen because you could write

( 0 ) (( 0 ) ) ( ( − 1 ) )
0 = A− 1 0 = A−1 A 1 =

( )( − 1 ) ( − 1 ) ( − 1 )
= A −1A 1 = I 1 = 1 ,

a contradiction. Thus the answer is that A does not have an inverse.

Example 18.6.5Let A =

( )
1 1
1 2

. Show

( )
2 − 1
− 1 1

is the inverse of A.

To check this, multiply

( ) ( ) ( )
1 1 2 − 1 = 1 0
1 2 − 1 1 0 1

and

( 2 − 1) ( 1 1 ) ( 1 0 )
− 1 1 1 2 = 0 1

showing that this matrix is indeed the inverse of A.