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For $n \geq 2$, compute the determinant of the following matrix: $$ B = \begin{bmatrix} -X & 1 & 0 & \cdots & 0 & 0 \\ 0 & -X & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 & \vdots \\ \vdots & & \ddots & \ddots & 1 & 0 \\ 0 & \cdots & \cdots & 0 & -X & 1 \\ a_0 & a_1 & \cdots & \cdots & a_{n-2} & (a_{n-1} - X) \end{bmatrix} $$

I am thinking of just multiplying the matrix along the diagonal but I'm not sure if that would work.

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    $\begingroup$ Laplace and induction? $\endgroup$ – user251257 Apr 6 '16 at 0:27
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    $\begingroup$ The first part is $-X\cdot \det(\text{sub-matrix})$ and the second part is $1\cdot \det(\text{sub-matrix of } 1)$ $\endgroup$ – Jared Apr 6 '16 at 0:30
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    $\begingroup$ Characteristic polynomial of a companion matrix $\endgroup$ – Robert Israel Apr 6 '16 at 0:30

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