Multiplication of Rational Matrices Let $\mathbf A(x)$ and $\mathbf B(x)$ be $n \times n$ rational matrices, whose elements are rational functions in the scalar $x \in \mathbb R$. Suppose that $\mathbf A(x) \mathbf B(x)$ is a polynomial matrix in $x$, meaning that the denominators in the elements of $\mathbf A(x)$ and $\mathbf B(x)$ somehow cancel out in the multiplication. 

Does it follow that $\mathbf B(x) \mathbf A(x)$ is also a polynomial matrix in $x$?

This seems like a simple question, but I haven't been able to prove it or come up with a counter example.

Here is my attempt at a counter example.
$$\left[\begin{array}{cc}
1&1/p(x)\\
0&1
\end{array}\right]\left[\begin{array}{cc}
1&1\\
p(x)&0
\end{array}\right] = \left[\begin{array}{cc}
2&1\\
p(x)&0
\end{array}\right]
$$
$$\left[\begin{array}{cc}
1&1\\
p(x)&0
\end{array}\right]\left[\begin{array}{cc}
1&1/p(x)\\
0&1
\end{array}\right]
=\left[\begin{array}{cc}
1&1+1/p(x)\\
p(x)&1
\end{array}\right]$$
