 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-1/1/b/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

Yes, in even dimension. For example, choose the skew-symmetric matrix

B = (01 - 1 0), C = (0 - 1 10) = - B .

Here

B^{2} = - I

, so

(I + B)^{- 1} = (I - B) /2

and

C = (I - B)^{2} /2 = - B

C = - C^{- 1}

gives

(I - B)^{2} = - (I + B)^{2}

, or

B^{2} = - I

; no real odd-dimensional matrix can satisfy that condition.

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