 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/18/a/solution

Let

e_{1}, e_{2}, e_{3}

be the Cartesian basis vectors, with the given vector

x = e_{1}

. The permutation matrix cycles them forward:

A x = 010, A^{2} x = 001, A^{3} x = 100 .

Thus three applications return the original vector.

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