5.1. MATRIX ARITHMETIC 93

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

Theorem 5.1.30 Suppose 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 tohave an inverse. This is illustrated in the following example.

Example 5.1.31 Let A =

(1 11 1

). Does A have an inverse?

One might think A would have an inverse because it does not equal zero. However,(1 11 1

)(−11

)=

(00

)

and if A−1 existed, this could not happen because you could write(00

)= A−1

((00

))= A−1

(A

(−11

))=

=(A−1A

)( −11

)= I

(−11

)=

(−11

),

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

Example 5.1.32 Let A =

(1 11 2

). Show

(2 −1−1 1

)is the inverse of A.

To check this, multiply(1 11 2

)(2 −1−1 1

)=

(1 00 1

)

and (2 −1−1 1

)(1 11 2

)=

(1 00 1

)showing that this matrix is indeed the inverse of A.