1
File: Notes 2 Elementary Row Operations 2009.doc
Elementary Row Operations (ERO)
An elementary row operation on a matrix
A
is one of the following operations
(
i
A
represents row
i
of matrix
A
):
(i)
Row
i
is interchanged with row
j
:
j
i
A
A
←⎯→
(ii)
Row
i
is multiplied by a nonzero scalar
k
:
i
i
kA
A
→
(iii)
Row
i
is replaced by
k
times row
j
+ row
i
:
j
i
i
kA
A
A
+
→
.
Elementary row operations are equivalent to pre-multiplying
A
by some matrix
B
.
For
example,
B
is obtained from an identity matrix with the following three variations for the
ERO’s:
(i): row (i) of
B
is
T
j
e
and row (j) of
B
is
T
i
e
;
(ii): row (i) of
B
is
T
i
ke
;
(iii): row (i) of
B
is
T
T
j
i
ke
e
+
.
The solution to a system of
m
equations in
n
unknowns,
Ax
b
=
, is invariant with respect
to a finite number of ERO’s.
That is
ˆ
x
solves
Ax
b
=
if and only if
ˆ
x
solves
A x
b
′
′
=
,
where
(
)
,
A b
′
′
can be obtained from
(
)
,
A b
by a finite number of
ERO’s.
Thus, there
exists a matrix
M
such that
.
Ax
b
MAx
M b
A x
b
=
=
′
′
=
For example, consider the matrix
2
1
1
1
2
1
1
1
2
A
⎡
⎤
⎢
⎥
=
−
⎢
⎥
−
⎢
⎥
⎣
⎦
.
To use (i) such as interchanging rows 1 and 3,
M
would be

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*