# Eigenvalues and Eigenvectors of a polynomial of matrix

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 $$M$$.

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.