 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014/8/solution

Here

M^{2} = - I

, so its even powers alternate between

\pm I

and its odd powers between

\pm M

. Collecting even and odd terms of the matrix exponential yields

R (ϕ) = I cos ϕ + M sin ϕ = (cos ϕ sin ϕ - sin ϕ cos ϕ) .

Thus

R^{T} R = I

det R = 1

, and

R (ϕ) (1, 0)^{T} = (cos ϕ, sin ϕ)^{T}

. It is the rotation matrix for a counterclockwise rotation through angle

ϕ

, preserving lengths and orientation.

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