4-number-matrix/Characteristic polynomial/Exercise

Let

${}M={\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\\-1&0&0&1\end{pmatrix}}\,.$

1. Determine the characteristic polynomial of ${}M$.
2. Determine a zero of the characteristic polynomial of ${}M$, and write the polynomial using the corresponding linear factor.
3. Show that the characteristic polynomial of ${}M$ has at least two real roots.

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