Let A and B be n x n matrices with A B Prove A and B has th

Let A and B be n x n matrices with A ~ B. Prove A and B has the same rank.

Solution

Let A and B be n x n matrices with A ~ B.That is A and B are two equivalent matrices.

Now we prove that A and B have same rank.

Since A ~ B , so that B can be obtained from A by applying a finite number of elementary transformations.

But elementary transformations do not alter the rank of a matrix.

hence rank(A) = rank(B)