Matrix multiplication interacts nicely with many operations. But not always with transforming a matrix to reduced row-echelon form. Suppose that $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Let $P$ be a matrix that is row-equivalent to $A$ and in reduced row-echelon form, $Q$ be a matrix that is row-equivalent to $B$ and in reduced row-echelon form, and let $R$ be a matrix that is row-equivalent to $AB$ and in reduced row-echelon form. Is $PQ=R$? (In other words, with nonstandard notation, is $\text{rref}(A)\text{rref}(B)=\text{rref}(AB)$?)

Construct a counterexample to show that, in general, this statement is false. Then find a large class of matrices where if $A$ and $B$ are in the class, then the statement is true.