- existence of a unique solution,
- existence of an infinite number of solutions, and
- no solution.
DEFINITION 2.4.1 (Consistent, Inconsistent)
A linear system is called CONSISTENT if it admits a solution
and is called INCONSISTENT if it admits no solution.
The question arises, as to whether there are conditions under which the linear system
is consistent.
The answer to this question is in the affirmative. To proceed further, we need
a few definitions and remarks.
Recall that the row reduced echelon form of a matrix is unique and therefore, the number of non-zero rows is a unique number. Also, note that the number of non-zero rows in either the row reduced form or the row reduced echelon form of a matrix are same.
DEFINITION 2.4.2 (Row rank of a Matrix)
The number of non-zero rows in the row reduced form of a matrix is called the
row-rank of the matrix.
By the very definition, it is clear that row-equivalent matrices
have the same row-rank. For a matrix
we write
`
' to denote the row-rank of
EXAMPLE 2.4.3
- Determine the row-rank of
Solution: To determine the row-rank of we proceed as follows. The last matrix in Step 1d is the row reduced form of which has non-zero rows. Thus, This result can also be easily deduced from the last matrix in Step 1b. - Determine the row-rank of
Solution: Here we have From the last matrix in Step 2b, we deduce
Remark 2.4.4
Let
be a linear system with
equations and
unknowns. Then the row-reduced echelon form of
agrees with the first
columns of
and hence
The reader is advised to supply a proof.
The reader is advised to supply a proof.
Remark 2.4.5
Consider a matrix
After application of a finite number
of elementary column operations (see
Definition 2.3.16) to the matrix
we can have a matrix, say
which has the following properties:
Therefore, we can define column-rank of
as the number
of non-zero columns in
It will be proved later that
- The first nonzero entry in each column is
- A column containing only 0 's comes after all columns with at least one non-zero entry.
- The first non-zero entry (the leading term) in each non-zero column moves down in successive columns.
Thus we are led to the following definition.
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