Page 69 - A Course in Linear Algebra with Applications

2.3: Elementary Matrices 53

(d) A is a product of elementary matrices.
Proof
We shall establish the logical implications (a) —> (b), (b) —•
(c), (c) —> (d), and (d) —> (a). This will serve to establish the
equivalence of the four statements.
If (a) holds, then A~ l exists; thus if we multiply both
1
sides of the equation AX = 0 on the left by A" , we get
1 1 - 1
A' AX = A^ 0, so that X = A 0 = 0 and the only solution
of the linear system is the trivial one. Thus (b) holds.
If (b) holds, then we know from 2.1.3 that the number of
pivots of A in reduced row echelon form is n. Since A is n x n,
this must mean that I n is the reduced row echelon form of A,
so that (c) holds.
If (c) holds, then 2.3.2 shows that there are elementary
matrices E±, ...,Ek such that Ek • • • E±A = I n. Since elemen-
tary matrices are invertible, E k- • -E\ is invertible, and thus
1
A=(E k--- Ei)" 1 = E^ 1 • • • E^ , so that (d) is true.
Finally, (d) implies (a) since a product of elementary ma-
trices is always invertible.

A procedure for finding the inverse of a matrix
As an application of the ideas in this section, we shall
describe an efficient method of computing the inverse of an
invertible matrix.
Suppose that A is an invertible n x n matrix. Then
there exist elementary n x n matrices E\, E^, • • •, E k such that
E k--- E 2E XA = I n, by 2.3.2 and 2.3.5. Therefore

1 l 1
A- = A~ = (£*••• E 2E lA)A~ = (E k--- E 2E 1)I n.
I n
This means that the row operations which reduce A to its
reduced row echelon form I n will automatically transform I n
- 1
to A . It is this crucial observation which enables us to
x
compute A~ .