Page 139 - Matrix Analysis & Applied Linear Algebra

3.9 ElementaryMatrices and Equivalence 133
Example 3.9.1
 
12 4
The sequence of row operations used to reduce A =  24 8  to E A is
3613
indicated below.
   
12 4 124
A =  24 8  R 2 − 2R 1 −→  000 
3613 R 3 − 3R 1 001
   
124 R 1 − 4R 2 120
Interchange R 2 and R 3
−−−−−−−−→  001  −→  001  = E A .
000 000
The reduction can be accomplished by a sequence of left-hand multiplications
with the corresponding elementary matrices as shown below.
       
1 −40 100 100 100
 0 1 0   001   010   −210  A = E A .
0 0 1 010 −301 001

 
13 0 −4
The product of these elementary matrices is P =  −30 1  , and you can
−21 0
verify that it is indeed the case that PA = E A . Thus the arrows are eliminated
by replacing them with a product of elementary matrices.

We are now in a position to understand why nonsingular matrices are pre-
cisely those matrices that can be factored as a product of elementary matrices.

Products of Elementary Matrices

• A is a nonsingular matrix if and only if A is the product (3.9.3)
of elementary matrices of Type I, II, or III.

Proof. If A is nonsingular, then the Gauss–Jordan technique reduces A to
I by row operations. If G 1 , G 2 ,..., G k is the sequence of elementary matrices
that corresponds to the elementary row operations used, then
G k ··· G 2 G 1 A = I or, equivalently, A = G −1 G −1 ··· G −1 .
1 2 k
Since the inverse of an elementary matrix is again an elementary matrix of the
same type, this proves that A is the product of elementary matrices of Type I,
II, or III. Conversely, if A = E 1 E 2 ··· E k is a product of elementary matrices,
then A must be nonsingular because the E i ’s are nonsingular, and a product
of nonsingular matrices is also nonsingular.