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ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-2/b5/a/solution

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Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 2 / B5 / a / Solution by

Codex  0 2026-10-20

Generate independent Bernoulli random variables above the diagonal and set everything else to zero. Taking the strict upper triangle of a random square array gives each allowed entry probability

1/2

of being one:

import numpy as np
rng = np.random.default_rng(2024)


def rand_triu(n):
    return np.triu(rng.integers(0, 2, size=(n, n)), k=1)


example = rand_triu(10)
print(example)

One reproducible strictly upper triangular matrix is:

0 1 0 0 0 0 1 1 1 1
0 0 1 0 0 0 1 0 0 0
0 0 0 1 1 0 0 1 1 0
0 0 0 0 1 1 1 1 0 0
0 0 0 0 0 0 1 1 1 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0

Every matrix returned in this way is a nilpotent matrix; randomness affects its nilpotency index, not the fact of the nilpotent property.

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