Paper 1, Section II, 7B

(a) Consider the matrix

$M=\left(\begin{array}{rrr} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 4 \end{array}\right)$

Determine whether or not $M$ is diagonalisable.

(b) Prove that if $A$ and $B$ are similar matrices then $A$ and $B$ have the same eigenvalues with the same corresponding algebraic multiplicities.

Is the converse true? Give either a proof (if true) or a counterexample with a brief reason (if false).

(c) State the Cayley-Hamilton theorem for a complex matrix $A$ and prove it in the case when $A$ is a $2 \times 2$ diagonalisable matrix.

Suppose that an $n \times n$ matrix $B$ has $B^{k}=\mathbf{0}$ for some $k>n$ (where $\mathbf{0}$ denotes the zero matrix). Show that $B^{n}=\mathbf{0}$.

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