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

{
1 if i = j
δij = 0 if i ⁄= j

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

Lemma 5.4.1Suppose A is an m×n matrix and I_{n}is the n×n identity matrix. Then AI_{n} = A.If I_{m}is the m × m identity matrix, it also follows that I_{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 5.4.2An n×n matrix A has an inverse, A^{−1}if and only if AA^{−1} = A^{−1}A = I. Sucha 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 5.4.3Suppose A^{−1}exists and AB = BA = I. Then B = A^{−1}.

Proof:

( )
A− 1 = A− 1I = A− 1(AB ) = A−1A 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 5.4.4Let 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 −1 0 −1 − 1
0 = A 0 = A A 1 =

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

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

Example 5.4.5Let 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.