Page 64 - A Course in Linear Algebra with Applications

48 Chapter Two: Systems of Linear Equations

and
E 5A = ( au ° 1 2 ") .
\ca 2i ca 22J
Thus premultiplication by E\ interchanges rows 1 and 2; pre-
multiplication by E 2 adds c times row 2 to row 1; premultipli-
cation by £5 multiplies row 2 by c . What then is the general
rule?
Theorem 2.3.1
Let A be an m x n matrix and let E be an elementary m xm
matrix.
(i) / E is of type (a), then EA is the matrix obtained
/
from A by interchanging rows i and j of A;
(ii) if E is type (b), then EA is the matrix obtained from
A by adding c times row j to row i;
(iii) if E is of type (c), then EA arises from A by multi-
plying row i by c.
Now recall from 2.2 that every matrix can be put in re-
duced row echelon form by applying elementary row opera-
tions. Combining this observation with 2.3.1, we obtain
Theorem 2.3.2
Let A be any mxn matrix. Then there exist elementary mxm
matrices E\, E 2, • ••, Ek such that the matrix EkE^-i • • • E\A
is in reduced row echelon form.

Example 2.3.2
Consider the matrix

A=
[2 1 oj-

We easily put this in reduced row echelon form B by applying
successively the row operations R\ <-> R 2, ^R\, R\ — ^R2 •

. (2 1 0 \ (I 1/2 0 \ (I 0 - 1 \ _
^ ^ 0 1 2)^\Q 1 2 ; ~ ^ 0 1 2J~