what happens to rank of matrices when singular matrix multiply by non-singular matrix?? I need to find the prove for any rectangular matrix $A$ and any non-singular matrices $P$, $Q$ of appropriate sizes,
rank($PAQ$)= rank($A$)
I know that when singular matrix multiply by non-singular the result will be singular matrix. However, It is not relevant to the rank of the matrix. If A is singular with rank 2, why rank($PAQ$)= 2 not any numbers.

Multiplying a matrix by a nonsingular matrix corresponds to composing some linear map (the possibly singular matrix) with an isomorphism (the invertible matrix). In particular an isomorphism (non-singular matrix) is essentially the identity map (i.e., the matrix is similar to the identity matrix), thus composing a linear map with an isomorphism preserves any linear structure the original map had to begin with, whether you compose from the left or the right.                                
