This problem is some variation of another MO question. Consider the matrix $$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c & \ddots & a \\ \vdots & \ddots & \ddots & \ddots & a \\ b & b & b & \dots & (-1)^nc \end{bmatrix}_{n\times n},$$ that is, a matrix with diagonal $(-1)^ic$, below diagonal entries $b$ and above diagonal entries $a$.

Question 1.Is there a nice closed formula for the determinant $\det(M_n)$? This has been solved.

Question 2.Is there a nice expression for the inverse $M_n^{-1}$? This one awaits an answer.

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