 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit/8/c/iii/solution

Yes. The old matrix is triangular, with determinant zero and trace

0 + 0 + 1 + 1 = 2

. The upper-left two-by-two block of the new matrix has determinant zero, so the whole determinant remains zero. Its trace is

- \frac{1}{2} + \frac{1}{2} + \frac{3}{2} + \frac{1}{2} = 2.

Thus

det [F]_{B} = det [F]_{C} = 0, tr [F]_{B} = tr [F]_{C} = 2.

This verifies both change of basis invariants from part (b).

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