Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 6A / a

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- Solution a

Solution

Using the convention of the question, the characteristic polynomial and characteristic equation are

χ_{A} (z) = det (A - z I), χ_{A} (z) = 0.

For

A = (a c b d),

one has

χ_{A} (z) = z^{2} - (a + d) z + (a d - b c) .

Direct multiplication gives

A^{2} - (a + d) A + (a d - b c) I = 0,

entry by entry. Thus

χ_{A} (A) = 0

, verifying the Cayley-Hamilton theorem for

2 \times 2

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