Eigenvalues and eigenspaces/-5 0 7 6 2 -6 -4 0 6/Exercise

Let

${}A={\begin{pmatrix}-5&0&7\\6&2&-6\\-4&0&6\end{pmatrix}}\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )\,.$

Compute:

1. the eigenvalues of ${}A$;
2. the corresponding eigenspaces;
3. the geometric and algebraic multiplicities of each eigenvalue;
4. a matrix ${}C\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )$ such that ${}C^{-1}AC$ is a diagonal matrix.

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