Question

Econometrics: Can someone please give a clear, concise and intuitive explanation of the rank of a matrix and how to find the rank using examples. WITHOUT REFERENCE TO ECHELON FORM.

Answer 1

Rank of a matrix:
EXPLANATION:

Rank of a matrix A is the maximum number of linearly independent rows of A, which is same as the maximum number of linearly independent columns of A.

EXAMPLES:

(1) To find the Rank of a 2 X 2 matrix:

We note:

(i)
Rank of Matrix A is 2 if:

Since both the column vectors are independent in this case.

(ii) Rank of Matrix A is 1 if

but

since both column vectors are not linearly independent, but there is a single column vector that is linearly independent, i.e., non-zero.

(iii) Rank of Matrix A is 0, if

(2) EXAMPLE OF FINDING RANK OF A MATRIX USING DETERMINANTS:

Given:
A matrix of order m X n.

A minor of matrix A of order k is a determinant of a k X k sub-matrix of A.

Computing rank:

Start with the minors of maximal order k.

If there is one that is non-zero, then, Rank =k.

If all maximal minors are 0, then Rank of A < k.

Then, we continue with the minors of order k - 1.

etc.

Example:

Consider matrix A given by:

The maximal minors have order = 3.

We find one obtained by deleting the last column = - 4 0.

Hence Rank of matrix A = 3