In previous sections, we solved linear systems using Gauss elimination method or the Gauss-Jordan method. In the examples considered, we have encountered three possibilities, namely

- existence of a unique solution,
- existence of an infinite number of solutions, and
- no solution.

Based on the above possibilities, we have the following definition.

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.

- 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

- 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.

Therefore, we can define **column-rank** of
as the number
of non-zero columns in
It will be proved later that

Thus we are led to the following definition.

We now apply column operations to the matrix Let be the matrix obtained from by successively interchanging the and column of for Then the matrix can be written in the form where is a matrix of appropriate size. As the block of is an identity matrix, the block can be made the zero matrix by application of column operations to This gives the required result. height6pt width 6pt depth 0pt

as the elementary martices 's are being multiplied on the left of the matrix Let be the columns of the matrix . Then check that for . Hence, we can use the 's which are non-zero (Use Exercise 1.2.17.2) to generate infinite number of solutions. height6pt width 6pt depth 0pt

- Determine the ranks of the coefficient and the augmented matrices that appear in Part 1 and Part 2 of Exercise 2.3.12.
- Let be an matrix with Then prove that is row-equivalent to
- If and are invertible matrices and is defined then show that
- Find matrices and which are product of elementary matrices such that where and
- Let
and
be two matrices. Show that
- if is defined, then
- if is defined, then and

- Let
be any matrix of rank
Then show that there
exists invertible matrices
such that

and Also, prove that the matrix is an invertible matrix. - Let be an matrix of rank Then can be written as where both and have rank and is a matrix of size and is a matrix of size
- Let
and
be two matrices such that
is defined and
Then
show that
for some matrix
Similarly, if
is
defined and
then
for some matrix
*[Hint: Choose non-singular matrices and such that and Define ]* - If matrices
and
are invertible
and the involved partitioned products are defined, then show
that
- Suppose
is the
inverse of a matrix
Partition
and
as follows:

A K Lal 2007-09-12