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
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:
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?