I'm having trouble understanding the proof of Theorem 1 on this page:

Pauls Online Notes : Linear Algebra - Finding Inverse Matrices

I don't understand how they proved the first conditional

If A in invertible, then this means that exists and that .

It seems that they assumed that , selected a solution to write , and then mulitplied both sides by so that

(Thinking)

I don't understand the logic here. It just looks like circular reasoning to me. Here is a quote from the proof:

So it seems that the equation that they are multiplying by is:Quote:

We start by assuming that is any solution to the system. Plug this into the system and then mulitply both sides by

But where does that come from? How is it that substituting into gives . If I look at it this way:

=

To get the right side to be equal to zero, they must have taken , but this isn't ANY solution. Can anyone help clarify this?