A First Course in Linear Algebra · Problem MINM.M80

Matrices — Robert Beezer

Mathematics — Linear Algebra · rref-matrix-multiplication

Problem

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.