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I am facing a problem where I'd appreciate some help:

It is $n \in \mathbb{N}$ and let $R$ be a commutative ring with $a_0,...,a_{n-1} \in R$. Show the following:
$\det(tI_n - A) =t^n - \sum\nolimits_{i=0}^{n-1}a_i t^i$ for $t \in R$ and

$A= \begin{pmatrix} 0 & & & & & a_0 \\ 1 & 0 & & & & a_1\\ & 1 & 0 & & & a_2 \\ & & \ddots & \ddots & & \vdots\\ & & & 1 & 0 & a_{n-2}\\ & & & & 1 & a_{n-1} \end{pmatrix}$

My idea was solving it through induction. The initial step works fine and I planned using the Laplace expansion of the determinant in the induction step to get the matrix to a level where I could apply the induction hypothesis, but it isn't working well. Therefore a hint would be gracious.

Thx in advance!

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    FWIW this is a duplicate of [Show the determinant of a matrix](http://math.stackexchange.com/questions/2124709/show-the-determinant-of-a-matrix). – 2017-02-02
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    HInt: Laplace expansion along the last column. No *real* induction is needed. – 2017-02-02
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    @johny hunter: i wrote an answer to your question in the link suggested by dxiv. Note however that your question is a duplicate of the one dxiv is refering to, so make sure next time you really search this site well on any results which might help you. Kind regards – 2017-02-02
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    I appreciate the advice, I will make sure to do so next time – 2017-02-02

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