VTRMC 2004 Problem 1

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&00&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&AB&C\end{pmatrix},\enspace N=\begin{pmatrix}I&BA&C\end{pmatrix}$$ where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$ , the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$ . Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$ ) implies $N$ is invertible? Justify your answer.