 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-4/9e/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The upper-left block has characteristic and minimal polynomial

(x^{2} + 1)^{2}

, while the lower-right block has characteristic and minimal polynomial

x^{2} + 1

. Hence the corresponding module is

R [x] / ((x^{2} + 1)^{2}) \oplus R [x] / (x^{2} + 1) .

In the power bases from part (ii), an explicit normal form is

010000001000000100 - 1 0 - 2 000 000001 0000 - 1 0,

the direct sum of the companion matrices of

(x^{2} + 1)^{2}

and

x^{2} + 1

 New to topics? Read the docs here!