Eigenvalues and eigenspaces/-4 6 6 0 2 0 -3 3 5/Exercise

Let

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

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.

Create a solution