 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/18/b/ii/solution

For each possibility in part (i), squaring the diagonal entries gives

M^{2} = I

. Equivalently, the linear transformation fixes a vector space basis after being applied twice. Therefore

M^{2} x y z = x y z .

Geometrically, reflecting twice in the same plane returns every vector to itself.

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