48 CHAPTER 2. LINEAR TRANSFORMATIONS

There is a special matrix called I and defined by

Iij = δij

where δij is the Kronecker symbol defined by

δij =

{1 if i = j

0 if i ̸= j

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

Lemma 2.1.21 Suppose A is an m× n matrix and In is the n× n identity matrix. ThenAIn = A. If Im is the m×m identity matrix, it also follows that ImA = A.

Proof:

(AIn)ij =∑k

Aikδkj

= Aij

and so AIn = A. The other case is left as an exercise for you.

Definition 2.1.22 An n × n matrix A has an inverse A−1 if and only if there exists amatrix, denoted as A−1 such that AA−1 = A−1A = I where I = (δij) for

δij ≡

{1 if i = j

0 if i ̸= j

Such a matrix is called invertible.

If it acts like an inverse, then it is the inverse. This is the message of the followingproposition.

Proposition 2.1.23 Suppose AB = BA = I. Then B = A−1.

Proof: From the definition B is an inverse for A. Could there be another one B′?

B′ = B′I = B′ (AB) = (B′A)B = IB = B.

Thus, the inverse, if it exists, is unique. ■

2.1.4 Finding The Inverse of a Matrix

A little later a formula is given for the inverse of a matrix. However, it is not a good wayto find the inverse for a matrix. There is a much easier way and it is this which is presentedhere. It is also important to note that not all matrices have inverses.

Example 2.1.24 Let 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 1

)(−1

1

)=

(0

0

)