First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort of mental block, or just convinced myself that I can't do it, but I find the LA bewildering and frustrating, and my professor has been no help at all. I have absolutely no problem with the "math" side of things. Once operations have been defined I can solve any matrix or vector problem I'm given, but when the book asks me to "prove" something I'm often at a loss, unless I can prove it merely by performing some operations that reduce to the problem to the thing I'm trying to prove. Anyway, here's an example of a proof that I am unable to figure out: 1. The problem statement, all variables and given/known data Show that if ad-bc=0, then the equation Ax=0 has more than one solution. Why does this imply that A is not invertible? [Hint: First, consider a=b=0. Then, if a and b are not both zero, consider the vector x=<-b,a> (sorry, I tried to latex this up, but I don't know enough about it. x is a column vector). 2. Relevant equations Theorems I suppose? The definition of the inverse matrix? 3. The attempt at a solution I actually have the solution manual, and it walks through the answer step by step and I do not understand the process by which it got to an answer. It says that if you follow the hint you get a nonzero solution. Okay, I don't quite follow why that matters. Then it says to set x2 to <-b, a> and -cb+da=0 which is a non-trivial solution? I don't understand this part either. Then it says that Ax=0 has more than one solution which means it's not invertible. I just don't follow the logic... or maybe logic in general. Is there a way to state this that's more "mathy" and less "proofy" that doesn't rely on my having to remember like 7 theorems?