Linear Algebra, Theory And Applications, 2012a

114 ROW OPERATIONS
of those columns to its left, and so forth. Thus by Lemma 4.2.3 if a pivot column occurs
as the j th column from the left, it follows that in the row reduced echelon form there will be
one of the e k as the j th column.
There are three choices for row operations at each step in the above theorem. A natural
question is whether the same row reduced echelon matrix always results in the end from
following the above algorithm applied in any way. The next corollary says this is the case.
Definition 4.3.4 Two matrices are said to be row equivalent if one can be obtained from
the other by a sequence of row operations.
Since every row operation can be obtained by multiplication on the left by an elementary
matrix and since each of these elementary matrices has an inverse which is also an elementary
matrix, it follows that row equivalence is a similarity relation. Thus one can classify matrices
according to which similarity class they are in. Later in the book, another more profound
way of classifying matrices will be presented.
It has been shown above that every matrix is row equivalent to one which is in row
reduced echelon form. Note
⎛
⎜
⎝
x 1
. .
x n
⎞
⎟
⎠ = x 1 e 1 + ···+ x n e n
so to say two column vectors are equal is to say they are the same linear combination of the
special vectors e j .
Corollary 4.3.5 The row reduced echelon form is unique. That is if B,C are two matrices
in row reduced echelon form and both are row equivalent to A, then B = C.
Proof: Suppose B and C are both row reduced echelon forms for the matrix A. Then
they clearly have the same zero columns since row operations leave zero columns unchanged.
If B has the sequence e 1 , e 2 , ··· , e r occurring for the first time in the positions, i 1 ,i 2 , ··· ,i r ,
the description of the row reduced echelon form means that each of these columns is not a
linear combination of the preceding columns. Therefore, by Lemma 4.2.3, the same is true of
the columns in positions i 1 ,i 2 , ··· ,i r for C. It follows from the description of the row reduced
echelon form, that e 1 , ··· , e r occur respectively for the first time in columns i 1 ,i 2 , ··· ,i r
for C. Thus B,C have the same columns in these positions. By Lemma 4.2.3, the other
columns in the two matrices are linear combinations, involving the same scalars, ofthe
columns in the i 1 , ··· ,i k position. Thus each column of B is identical to the corresponding
column in C.
The above corollary shows that you can determine whether two matrices are row equivalent
by simply checking their row reduced echelon forms. The matrices are row equivalent
if and only if they have the same row reduced echelon form.
The following corollary follows.
Corollary 4.3.6 Let A be an m × n matrix and let R denote the row reduced echelon form
obtained from A by row operations. Then there exists a sequence of elementary matrices,
E 1 , ··· ,E p such that
(E p E p−1 ···E 1 ) A = R.
Proof: This follows from the fact that row operations are equivalent to multiplication
on the left by an elementary matrix.