Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 5 / e

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- Solution e

Solution

Here

A^{2} = - I

. Splitting the matrix exponential into even and odd powers therefore gives

U = I cos 1 + A sin 1 = (cos 1 - sin 1 sin 1 cos 1) .

For the second matrix,

(I + A) (I - A) = I - A^{2} = 2 I

, so

(I + A)^{- 1} = (I - A) /2

. Hence

V = \frac{1}{2} (I - A)^{2} = \frac{1}{2} (I - 2 A + A^{2}) = - A = (01 - 1 0) .

Both are real orthogonal matrices, and hence also unitary matrices.

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