I never took linear algebra, and I'm studying it on my own for the sake of learning (MIT opencourseware, online books, etc).
I'm currently working on that a matrix A is invertible, and I can show that an equivalent statement to this is that A is row equivalent to the n x n Identity matrix. Now, I'd like to show that the n x n identity matrix has n pivot positions. How would I get that started? I'm not looking for a formal proof, but an informal proof with justifications (doesn't have to be rigorous, just the sketch of the idea).