## An Example of Matrices $A$, $B$ such that $\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B)$

## Problem 569

For an $m\times n$ matrix $A$, we denote by $\mathrm{rref}(A)$ the matrix in reduced row echelon form that is row equivalent to $A$.

For example, consider the matrix $A=\begin{bmatrix}

1 & 1 & 1 \\

0 &2 &2

\end{bmatrix}$

Then we have

\[A=\begin{bmatrix}

1 & 1 & 1 \\

0 &2 &2

\end{bmatrix}

\xrightarrow{\frac{1}{2}R_2}

\begin{bmatrix}

1 & 1 & 1 \\

0 &1 & 1

\end{bmatrix}

\xrightarrow{R_1-R_2}

\begin{bmatrix}

1 & 0 & 0 \\

0 &1 &1

\end{bmatrix}\]
and the last matrix is in reduced row echelon form.

Hence $\mathrm{rref}(A)=\begin{bmatrix}

1 & 0 & 0 \\

0 &1 &1

\end{bmatrix}$.

Find an example of matrices $A$ and $B$ such that

\[\mathrm{rref}(AB)\neq \mathrm{rref}(A) \mathrm{rref}(B).\]