(a) Find a matrix $M$ over $\mathbb{C}$ with both minimal polynomial and characteristic polynomial equal to $(x-2)^{3}(x+1)^{2}$. Furthermore find two matrices $M_{1}$ and $M_{2}$ over $\mathbb{C}$ which have the same characteristic polynomial, $(x-3)^{5}(x-1)^{2}$, and the same minimal polynomial, $(x-3)^{2}(x-1)^{2}$, but which are not conjugate to one another. Is it possible to find a third such matrix, $M_{3}$, neither conjugate to $M_{1}$ nor to $M_{2}$ ? Justify your answer.

(b) Suppose $A$ is an $n \times n$ matrix over $\mathbb{R}$ which has minimal polynomial of the form $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)$ for distinct roots $\lambda_{1} \neq \lambda_{2}$ in $\mathbb{R}$. Show that the vector space $V=\mathbb{R}^{n}$ on which $A$ defines an endomorphism $\alpha: V \rightarrow V$ decomposes as a direct sum into $V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right)$, where $\iota$ is the identity.

[Hint: Express $v \in V$ in terms of $\left(\alpha-\lambda_{1} \iota\right)(v)$ and $\left.\left(\alpha-\lambda_{2} \iota\right)(v) .\right]$

Now suppose that $A$ has minimal polynomial $\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right) \ldots\left(x-\lambda_{m}\right)$ for distinct $\lambda_{1}, \ldots, \lambda_{m} \in \mathbb{R}$. By induction or otherwise show that

$$V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right) \oplus \ldots \oplus \operatorname{ker}\left(\alpha-\lambda_{m} \iota\right)$$

Use this last statement to prove that an arbitrary matrix $A \in M_{n \times n}(\mathbb{R})$ is diagonalizable if and only if all roots of its minimal polynomial lie in $\mathbb{R}$ and have multiplicity $1 .$