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probability of A^2=O after perturbation

Source: 2021 Simon Marais, B1

November 2, 2021
probabilitylinear algebramatrix

Problem Statement

Let n≥2n \ge 2 be an integer, and let OO be the n×nn \times n matrix whose entries are all equal to 00. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from 00 to 11. Call the resulting matrix AA. Determine the probability that A2=OA^2 = O, as a function of nn.
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