Kenneth Kuttler

390 CHAPTER 18. LINEAR FUNCTIONS

The row reduced echelon form is

 1 0 1 00 1 1 00 0 0 1

 and so the pivot columns are the

first, second, and last. Thus every column of A is a linear combination of these pivotcolumns.

18.2.3 The Inverse

Definition 18.2.25 Let A be an n×n matrix. It is said to be invertible if there is amatrix B such that AB = BA = I. Then B is called the inverse of A and is denoted by A−1.

First of all, the inverse, if it exists, is unique. To see this suppose both B, B̂ work in theabove definition. Then

B̂ = B̂I = B̂(AB) =(B̂A)

B = IB = B

This means that to show something is the inverse, it suffices to show that it acts like theinverse. If it walks like a duck and quacks like a duck, then it is a duck. However, althoughthere are many ducks, a given matrix has at most one inverse.

Recall the elementary matrices, how if E is one of them, there is another elementarymatrix of the same sort Ê such that ÊE = EÊ = I. This was Theorem 18.2.11 above. ThusÊ = E−1.

Lemma 18.2.26 A product of elementary matrices E1E2 · · ·En has an inverse and itsinverse is ÊnÊn−1 · · · Ê1, the product of the inverses in the reverse order.

Proof: In case n = 1, this was shown above. Suppose it is true for n matrices. Then

E1E2 · · ·EnEn+1Ên+1ÊnÊn−1 · · · Ê1

= E1E2 · · ·En(En+1Ên+1

)ÊnÊn−1 · · · Ê1

= E1E2 · · ·EnIÊnÊn−1 · · · Ê1 = E1E2 · · ·EnÊnÊn−1 · · · Ê1

and this is I by induction. It is exactly similar in the other order.

Ên+1ÊnÊn−1 · · · Ê1E1E2 · · ·EnEn+1

= Ên+1ÊnÊn−1 · · ·(Ê1E1

)E2 · · ·EnEn+1

= Ên+1ÊnÊn−1 · · · Ê2E2 · · ·EnEn+1 = I

by induction, since there are now only n matrices in each of the two products.Now here is the main result about inverses.

Theorem 18.2.27 Let A be an n×n matrix. Then A is invertible if and only if therow reduced echelon form of A is I. In this case A equals a finite product of elementarymatrices.

Proof: ⇐ Suppose the row reduced echelon form of A is I. Then, as shown above, thereare elementary matrices E1, · · · ,Em such that E1 · · ·EmA = I. Then, by Lemma 18.2.26,A = Êm · · · Ê1I = Êm · · · Ê1 and so A is the product of elementary matrices. By Lemma18.2.26 again, A−1 exists and equals E1 · · ·Em.