 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/3/a/solution

τ_{1}^{2} = τ_{2}^{2} = τ_{3}^{2} = I_{2} .

For example, the diagonal entries of

τ_{2}^{2}

are

(- i) i = i (- i) = 1

, and its off-diagonal entries vanish. Associativity now reduces every positive integer power to its integer parity:

τ_{j}^{2 m} = I_{2}, τ_{j}^{2 m + 1} = τ_{j} .

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