= Solution
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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