Given two square matrices $A,\,B$ of order $n$, of which therefore all the terms are known, let us define the following polynomial:
$$ p(x) := \det(A - x\,B)\,. $$
I was wondering if in the literature there was an algorithm to iteratively determine the coefficients of this polynomial starting from the terms of the two matrices. I tried to search before asking but couldn't find anything. Any advice is welcome.
In the case where $B=I$, according to the Faddeev–LeVerrier algorithm, we have:
$$ k = 0: \quad \quad \quad M_0 = \mathbf{0}\,, \quad \quad \quad c_n = (-1)^n\,; $$
$$ 1 \le k \le n: \quad \quad \quad M_k = A\cdot M_{k-1} + c_{n-k+1}\,I\,, \quad \quad \quad c_{n-k} = -\frac{\text{tr}(A\cdot M_k)}{k}\,. $$
If $B \ne I$ and $\exists\,B^{-1}$, then $A \mapsto A \cdot B^{-1}$ and in the end just multiply everything by $\det(B)$.
But if $\not\exists\,B^{-1}$, i.e. $\det(B)=0$, is there any chance?