 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/10/a/solution

The explicit matrices are

W = (1001), X = (i 0 0 - i), Y = (0 - 1 10), Z = (0 i i 0) .

WW = W, XX = YY = ZZ = W, X Y = Z, Y Z = X, ZX = Y .

The reverse products have the opposite signs:

Y X = Z

Z Y = X

XZ = Y

. Since

W

X^{2} = Y^{2} = Z^{2} = - W

W^{- 1} = W, X^{- 1} = X, Y^{- 1} = Y, Z^{- 1} = Z .

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