Linear Mathematics | Part IB, 2002

Define f:C3C3f: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3} by

f(a,b,c)=(a+3bc,2b+c,4bc)f(a, b, c)=(a+3 b-c, 2 b+c,-4 b-c)

Find the characteristic polynomial and the minimal polynomial of ff. Is ff diagonalisable? Are ff and f2f^{2} linearly independent endomorphisms of C3\mathbb{C}^{3} ? Justify your answers.

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