Chapter 8
Rank Of A Matrix8.1 Elementary Matrices
The elementary matrices result from doing a row operation to the identity matrix.
Definition 8.1.1 The row operations consist of the following
1. Switch two rows.
2. Multiply a row by a nonzero number.
3. Replace a row by a multiple of another row added to it.
The elementary matrices are given in the following definition.
Definition 8.1.2 The elementary matrices consist of those matrices which result by ap-plying a row operation to an identity matrix. Those which involve switching rows of theidentity are called permutation matrices1.
As an example of why these elementary matrices are interesting, consider the following. 0 1 01 0 00 0 1
a b c d
x y z wf g h i
=
x y z wa b c df g h i
A 3×4 matrix was multiplied on the left by an elementary matrix which was obtained fromrow operation 1 applied to the identity matrix. This resulted in applying the operation 1 tothe given matrix. This is what happens in general.
When you multiply by the identity, nothing happens, but when you multiply by anelementary matrix you end up doing a row operation. The next definition is what is meantby an elementary matrix.
Definition 8.1.3 The elementary matrices consist of those matrices which result by ap-plying a row operation to an identity matrix. Those which involve switching rows of theidentity are called permutation matrices2.
The importance of elementary matrices is that when you multiply on the left by one, itdoes the row operation which was used to produce the elementary matrix.
Now consider what these elementary matrices look like. First consider the one whichinvolves switching row i and row j where i < j. This matrix is of the form
. . .
0 1. . .
1 0. . .
1More generally, a permutation matrix is a matrix which comes by permuting the rows of the identity matrix,
not just switching two rows.2More generally, a permutation matrix is a matrix which comes by permuting the rows of the identity matrix,
which means possibly more than two rows are switched.
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