8.6. THE INVERSE OF A MATRIX 147
and row reduced till you obtained(1 0 | 2 −10 1 | −1 1
)and read off the inverse as the 2×2 matrix on the right side.
This is the reason for the following simple procedure for finding the inverse of a matrix.This procedure is called the Gauss-Jordan procedure.
PROCEDURE 8.6.6 Suppose A is an n×n matrix. To find A−1 if it exists, formthe augmented n×2n matrix
(A|I)and then, if possible do row operations until you obtain an n×2n matrix of the form
(I|B) . (8.18)
When this has been done, B = A−1. If it is impossible to row reduce to a matrix of the form(I|B) , then A has no inverse.
Actually, all this shows is how to find a right inverse if it exists. What has been shownfrom the above discussion is that AB = I. Later, I will show that this right inverse is theinverse. See Corollary 28.1.15 presented later. However, it is not hard to see that this shouldbe the case as follows.
The row operations are all reversible. If the row operation involves switching two rows,the reverse row operation involves switching them again to get back to where you started.If the row operation involves multiplying a row by a ΜΈ= 0, then you would get back to whereyou began by multiplying the row by 1/a. The third row operation involving addition of ctimes row i to row j can be reversed by adding −c times row i to row j.
In the above procedure, a sequence of row operations applied to I yields B while thesame sequence of operations applied to A yields I. Therefore, the sequence of reverse rowoperations in the opposite order applied to B will yield I and applied to I will yield A. Thatis, there are row operations which provide
(B|I)→ (I|A)
and as just explained, A must be a right inverse for B. Therefore, BA = I. Hence B is botha right and a left inverse for A because AB = BA = I.
If it is impossible to row reduce (A|I) to get (I|B) , then in particular, it is impossible torow reduce A to I and consequently impossible to do a sequence of row operations to I andget A. Later it will be made clear that the only way this can happen is that it is possible to
row reduce A to a matrix of the form
(C0
)where 0 is a row of zeros. Then there will be
no solution to the system of equations represented by the augmented matrix(C0| 0
1
)Using the reverse row operations in the opposite order on both matrices in the above, itfollows that there must exist a such that there is no solution to the system of equationsrepresented by (A|a). Hence A fails to have an inverse, because if it did, then there wouldbe a solution x to the equation Ax= a given by A−1a.