Paper 1, Section II, 7B

Vectors and Matrices | Part IA, 2012

(a) Consider the matrix

M=(210011024)M=\left(\begin{array}{rrr} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 4 \end{array}\right)

Determine whether or not MM is diagonalisable.

(b) Prove that if AA and BB are similar matrices then AA and BB 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 AA and prove it in the case when AA is a 2×22 \times 2 diagonalisable matrix.

Suppose that an n×nn \times n matrix BB has Bk=0B^{k}=\mathbf{0} for some k>nk>n (where 0\mathbf{0} denotes the zero matrix). Show that Bn=0B^{n}=\mathbf{0}.

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