Computing a few Invertible matrices

Problem: Enumerate and generate the number of invertible matrices (n*n) with each entry either 0 or 1.

Example: For n=2, we have a set of 6 matrices. They are

Interestingly, these matrices form a group under matrix multiplication.

A much more harder problem would be to enumerate and generate all invertible (n*n) matrices with enrites fron the set { 0,1,...,n-1 }.

Linear Independence

This problem is the generalization of exercise 2.3.9 from 'linear algebra' Hoffman and Kunze.

Let the set of vectors be linearly independent. Prove that are also linearly independent given



I mean


and so on.

ps: I proved it by proving a certain matrix to be invertible and that was pretty cumbersome. Hoping for a better pretty solution :)