Tag Archives: binary vectors

Random matrices and pentagonal numbers

Here is a clever problem:

Let us fix n. Take at random, with uniform probability, a n \times n binary matrix (call it A).
Let: \gamma_n = P[\det_{Z_2}(A)=1].
(You can think of it as taking determinant of matrix normally and asking for a probability that it is odd number.)
Find the value of: \gamma = \lim_{n\to\infty} \gamma_n

(It’s not that gamma! Although, it is still related to Euler!)
Continue reading