18.2. ROW OPERATIONS AND LINEAR EQUATIONS 391

⇒ Suppose now that A is invertible. Either every column of A is a pivot column,in which case the row reduced echelon form of A, called R, is the identity or else somecolumn is not a pivot column and in this case, R has a bottom row of zeros. I need torule out this case. However, since the bottom row of R is all zeros, there is no solutionx to Rx= en. Say E1 · · ·EmA = R where the Ei are elementary matrices correspondingto row operations which produced R. Then A = Êm · · · Ê1R and so there is no solutionto Ax = Êm · · · Ê1R x= Êm · · · Ê1b≡ c because by Proposition 18.2.12, multiplying bothsides of an equation by an elementary matrix preserves the solution set. Now this is acontradiction because if A−1 exists, then you would get a unique solution to Ax= c, namelyx= A−1c so the first case must hold that the row reduced echelon form of A is I.

Now it is not hard to give a simple algorithm for finding the inverse of an n×n matrixwhen it exists and to determine that there is no inverse in case it does not exist. From theabove, there are elementary matrices Ei such that

E1 · · ·EmA = R

where R is in row reduced echelon form. If R ̸= I, then there is no inverse. If R = I, thenthe inverse of A is E1 · · ·Em = E1 · · ·EmI. Thus you do a sequence of row operations to Iwhich gives the inverse of A with the same sequence of operations applied to A yielding I.This is summarized in the procedure for finding the inverse.

Procedure 18.2.28 Let A be an n×n matrix. Write (A|I) . Then do row operationsuntil you get the row reduced echelon form. If you get I on the left, then what remains onthe right will be the inverse of A. If you have a row of zeros on the left so the row reducedechelon form of A is not I, then A−1 does not exist.

Example 18.2.29 Find A−1 where A =

 4 1 10 2 −11 −1 1

 .

Write  4 1 1 1 0 00 2 −1 0 1 01 −1 1 0 0 1

and do row operations to find the row reduced echelon form. This yields 1 0 0 1 −2 −3

0 1 0 −1 3 40 0 1 −2 5 8

Now the inverse is what is on the right.

A−1 =

 1 −2 −3−1 3 4−2 5 8

You should always check your work. 1 −2 −3

−1 3 4−2 5 8

 4 1 10 2 −11 −1 1

=

 1 0 00 1 00 0 1

By the above discussion involving elementary matrices, a matrix obtained in this way

which acts like the inverse on the left will also act like the inverse on the right so it sufficesto check multiplication on only one side.