Eigenvalues and Eigenvectors of a polynomial of matrix

Question

Let $M$ be a $3\times 3$ matrix with eigenvalues $0,2,3$. What is then number of linearly independent eigenvectors of $M^3+2*M+I$?

I know that eigenvectors of distinct eigenvalues are linearly independent but I am stuck at finding eigenvalues of the above expression of $M$.

Answer 1

Since $0$, $2$, and $3$ are eigenvalues of $M$, then $1(=0^3+2\times0+1)$, $13(=2^3+2\times2+1)$ and $34(=3^3+2\times3+1)$ are eigenvalues of $M^3+2M+\operatorname{Id}$. Since that matrix is a $3\times3$ matrix with $3$ distinct eigenvalues, it has $3$ linearly independent eigenvectors.