Math 313 Lecture #6
§
1.5: Elementary Matrices, Part I
Obtaining Equivalent Systems by Matrix Multiplication.
Consider a linear
system
A~x
=
~
b
for
A
an
m
×
n
matrix.
If
M
is a nonsingular
m
×
m
matrix, then
A~x
=
~
b
is equivalent to
MA~x
=
M
~
b.
Why? Because if
~x
is a solution of
A~x
=
~
b
, then
~x
is a solution of
MA~x
=
M
~
b
; on the
other hand if
~x
is a solution of
MA~x
=
M
~
b
, then applying the inverse of
M
to both sides
gives
A~x
=
M

1
MA~x
=
M

1
M
~
b
=
~
b.
The idea is to choose
M
so that
MA~x
=
M
~
b
is easier to solve than
A~x
=
~
b
, i.e., ﬁnd
M
so that
MA
is in row echelon form.
Such an
M
might be found by applying a sequence of “simple enough” nonsingular
matrices
E
1
,...,E
k
to both sides of
A~x
=
~
b
:
E
k
···
E
1
A~x
=
E
k
···
E
1
~
b,
that is,
M
=
E
k
···
E
1
which is nonsingular because each
E
i
is.
Well, we know how to use row operations to get a row echelon form.
Can each row operation be achieved by matrix multiplication?
Examples.
Let
A
=
1 2 3
4 5 6
7 8 9
, and consider the following three matrices obtained
from the identity matrix by a single elementary row operation:
1 0 0
0 1 0
0 0 1
R
2
↔
R
3
⇔
1 0 0
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 Fall '10
 na
 Linear Algebra, Algebra, Multiplication, Matrices, ax, Invertible matrix

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