TITLE: Random Binary matrix
QUESTION [5 upvotes]: This is a question from Strang's "Linear Algebra and its Applications", right in the first chapter (I'm studying it by myself). I couldn't solve it, it isn't in the Solutions Manual, and my research suggests that there shouldn't be a simple solution for it. However, its presence in the very first chapter suggests me that I'm missing something. Here it goes:
1.6:
a) There are sixteen 2x2 matrices whose entries are 1's and 0's. How many are invertible?
b) (Much harder!) If you put 1's and 0's at random into the entries of a 10 by 10 matrix, is it more likely to be invertible or singular?
From what I can tell, this can't be solved by elementary linear algebra so... it shouldn't be there? I'm guessing there is a clever computational way to exhaust all cases?

REPLY [3 votes]: @FrancoVS @N74 This is not a proof, rather an indication that the larger the matrix, the more the odds that it IS invertible.  The broken line below displays the results of a (Matlab) simulation:  for each $n = 1,2 \cdots 20$, we have generated $20,000$ $n \times n$ matrices (with entries following a Bernoulli Ber(1/2) distribution), and their determinant has been computed. This graphic displays the fact that, for $n>15$, the frequency of non-zero determinants rapidly tends to 1... (after a small decline for small values of $n$).