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:
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?