Strategy to Solve Proof Questions

Hi, I'm not sure how much help you guys can give, since this isn't really a textbook question...More of a "how should I study this".

I find I'm able to do calculations and the like with ease in my linear algebra course, but it's the proofs that really get me. Every time I see a question that says "Show that..." or "Prove that..." I'm not able to figure out a method to complete the question. If I see the answer, I can understand just fine, but I don't understand how any person can think of that method to solve it.

For example, one of the midterm questions I received:

"If A and B are n by n matrices, and given that AB is invertible, prove that B must be invertible."

I draw a blank when I try to solve this question on my own.

When the solutions came up, I understood perfectly. To prove B is invertible, Bx = 0 must only have the trivial solution of x = 0. And then you start off with (AB)x = 0 and can easily go from there.

So, are there some tips or strategies to get better at doing these questions?