[Math] what happens to rank of matrices when singular matrix multiply by non-singular matrix?

Question

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.

Answer 1

A square non-singular (or invertible) is row equivalent to the identity matrix (with the same number of rows/columns). Thus, $P$ and $Q$ can be written as the product of a finite number of elementary matrices.

Thus $PAQ$ can be formed from $A$ by performing row operations and column operations. These operations preserve the rank of $A$.