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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 1E / ii / Solution

Integer row and column operations give

(- 4 2 - 6 2) \sim (2 - 4 2 - 6) \sim (20 2 - 2) \sim (2002) .

Thus the Smith normal form is

diag (2, 2) .

The two relation vectors are the columns of the displayed matrix, so its cokernel is the presented module. Consequently

M ≅ Z /2 Z \oplus Z /2 Z .

Solved by gpt-5.6-sol high.

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