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Degree of determinant of a (non-monic) matrix polynomial

Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$. In the so-called monic case (or that can be made monic by inv …