Linear Algebra, Theory And Applications, 2012a

48 MATRICES AND LINEAR TRANSFORMATIONS
If it acts like an inverse, then it is the inverse.
proposition.
Proposition 2.1.23 Suppose AB = BA = I. Then B = A −1 .
This is the message of the following
Proof: From the definition B is an inverse for A. Could there be another one B ′ ?
Thus, the inverse, if it exists, is unique.
B ′ = B ′ I = B ′ (AB) =(B ′ A) B = IB = B.
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 way
to find the inverse for a matrix. There is a much easier way and it is this which is presented
here. 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 0
=
1 1 1 0
and if A −1 existed, this could not happen because you could multiply on the left by the
inverse A and conclude the vector (−1, 1) T =(0, 0) T . Thus the answer is that A does not
have an inverse.
Suppose you want to find B such that AB = I. Let
B = ( )
b 1 ··· b n
Also the i th column of I is
e i = ( 0 ··· 0 1 0 ··· 0 ) T
Thus, if AB = I, b i , the i th column of B must satisfy the equation Ab i = e i . The augmented
matrix for finding b i is (A|e i ) . Thus, by doing row operations till A becomes I, you end up
with (I|b i )whereb i is the solution to Ab i = e i . Now the same sequence of row operations
works regardless of the right side of the agumented matrix (A|e i ) and so you can save trouble
by simply doing the following.
(A|I)
row operations
→
(I|B)
and the i th column of B is b i , the solution to Ab i = e i .ThusAB = I.
This is the reason for the following simple procedure for finding the inverse of a matrix.
This procedure is called the Gauss Jordan procedure. It produces the inverse if the matrix
has one. Actually, it produces the right inverse.
Procedure 2.1.25 Suppose A is an n × n matrix. To find A −1 if it exists, form the
augmented n × 2n matrix,
(A|I)
and then do row operations until you obtain an n × 2n matrix of the form
(I|B) (2.18)
if possible. When this has been done, B = A −1 . The matrix A has an inverse exactly when
it is possible to do row operations and end up with one like (2.18).