I have a polynomial matrix $M(x) = f(x) I + g(x)A + AB$, where $f(x), g(x)$ are polynomials of degree $k, l$ respectively, $I$ is $N\times N$ identity matrix and $A, B$ some $N\times N$ matrices. Are there any polynomials $h(x), p(x)$ and $2N \times 2N$ matrix $C$ such that matrix polynomial $h(x) I_{2N\times 2N} + p(x) C$ has the same roots as $M(x)$, i.e. $\det(M(x))$ and $\det(h(x) I_{2N\times 2N} + p(x) C)$ have the same roots?

**Motivation**

For the second-order system the matrix polynomial $Q(x) = Mx^2 + K$ has the same roots as $xI_{2N\times 2N} + \begin{bmatrix} 0 & -I\\ M^{-1} K & 0 \end{bmatrix}$ corresponding to the linear matrix equation. The question above is the extension of this 'trick'.