How can I calculate the determinant of the following matrix?
$$A=\begin{pmatrix} \lambda&&&a_0\\ -1&\ddots&&\vdots\\ &\ddots&\lambda&a_{n-2}\\ &&-1&\lambda+a_{n-1} \end{pmatrix}$$
I've tried to calculate the determinants of the $2 \times 2$ and $3 \times 3$ ones and finally got something like
$$\det(A_n)=\lambda^n+\lambda^{n-1}a_{n-1}+\dots+a_0$$
But in order to prove this solution a mathematical induction would be necessary. Is there any more efficient way?
