 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/17/c/ii/solution

Tr Y = 2 α, det Y = α^{2} + 2 β^{2} .

Since

Tr X = 0

and

det X = 2

, the extra conditions force

α = 0

and

β^{2} = 1

. Hence

Y = X or Y = - X .

Both choices are real, commute with

X

, and satisfy the required trace and determinant conditions.

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