Rank of a Matrix

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

1. existence of a unique solution,
2. existence of an infinite number of solutions, and
3. no solution.

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

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
1. 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.
2. 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

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:
1. The first nonzero entry in each column is
2. A column containing only 0 's comes after all columns with at least one non-zero entry.
3. 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.

DEFINITION 2.4.6   The number of non-zero rows in the row reduced form of a matrix is called the rank of denoted

THEOREM 2.4.7   Let be a matrix of rank Then there exist elementary matrices and such that

Proof. Let be the row reduced echelon matrix obtained by applying elementary row operations to the given matrix As the matrix will have the first rows as the non-zero rows. So by Remark 2.3.5, will have leading columns, say Note that, for the column will have in the row and zero elsewhere.

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

COROLLARY 2.4.8   Let be a matrix of rank Then the system of equations has infinite number of solutions.

Proof. By Theorem 2.4.7, there exist elementary matrices and such that Define . Then the matrix

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

EXERCISE 2.4.9
1. Determine the ranks of the coefficient and the augmented matrices that appear in Part 1 and Part 2 of Exercise 2.3.12.
2. Let be an matrix with Then prove that is row-equivalent to
3. If and are invertible matrices and is defined then show that
4. Find matrices and which are product of elementary matrices such that where and
5. Let and be two matrices. Show that
1. if is defined, then
2. if is defined, then and
6. 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.
7. 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
8. 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 ]
9. If matrices and are invertible and the involved partitioned products are defined, then show that

10. Suppose is the inverse of a matrix Partition and as follows:

If is invertible and then show that

and

A K Lal 2007-09-12