(a) Find the eigenvalues and eigenvectors of the matrix

$$M=\left(\begin{array}{rrr} 2 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & -2 & 3 \end{array}\right)$$

(b) Under what conditions on the $3 \times 3$ matrix $A$ and the vector $\mathbf{b}$ in $\mathbb{R}^{3}$ does the equation

$$A \mathbf{x}=\mathbf{b}$$

have 0,1 , or infinitely many solutions for the vector $\mathbf{x}$ in $\mathbb{R}^{3}$ ? Give clear, concise arguments to support your answer, explaining why just these three possibilities are allowed.

(c) Using the results of $(\mathrm{a})$, or otherwise, find all solutions to $(*)$ when

$$A=M-\lambda I \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{l} 4 \\ 3 \\ 2 \end{array}\right)$$

in each of the cases $\lambda=0,1,2$.