 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/1/b/ii/solution

Write

A = I_{3} + N

, where

N = 000 a 00 b c 0, N^{2} = 000000 a c 00, N^{3} = 0.

I_{3} N = N I_{3}

. All terms of degree at least three vanish, so for every positive integer

n

A^{n} = I_{3} + n N + (2 n) N^{2} = 100 na 10 nb + \frac{n ( n - 1 )}{2} a c n c 1 .

The same expression directly gives

A

when

n = 1

, so no small exponent is excluded.

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