4-number-matrix/Zero/Eigenvector/Exercise

Let ${}\lambda$ be a zero of the polynomial

$X^{3}+2X^{2}-2.$

Show that

${\begin{pmatrix}{\frac {1}{(1+\lambda )^{3}}}\\{\frac {1}{(1+\lambda )^{2}}}\\{\frac {1}{(1+\lambda )}}\\1\end{pmatrix}}$

is an eigenvector of the matrix

${\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\\-1&0&0&1\end{pmatrix}}$

for the eigenvalue ${}\lambda$.

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