(a) Determine the characteristic polynomial and the eigenvectors of the matrix

$$\left(\begin{array}{ccc} 0 & 1 & 0 \\ -1 & 2 & 0 \\ 1 & -1 & 2 \end{array}\right)$$

Is it diagonalizable?

(b) Show that an $n \times n$ matrix $A$ with characteristic polynomial $f(t)=(t-\mu)^{n}$ is diagonalizable if and only if $A=\mu I$.