8.6. THE INVERSE OF A MATRIX 145

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 8.6.4 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 8.6.5 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.

8.6.2 Finding The Inverse Of A Matrix

In the last example, how would you find A−1? You wish to find a matrix

(x zy w

)such

that (1 11 2

)(x zy w

)=

(1 00 1

).

This requires the solution of the systems of equations,

x+ y = 1,x+2y = 0