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 × n matrix and B is an n × 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 rref(A)rref(B) = 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.