Matrix inverses - Why are they derived the way they are? Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process.
There are questions on SE about proofs for the inverse, but niether here nor anywhere else on the internet, can I find a decent explanation of why it works the way it does:
So, basically, suppose the matrix is $\begin{bmatrix} a && b \\ c && d \end{bmatrix}$
Why do we swap values $a$ and $d$?
Why do $c$ and $b$ become negative?.
Why do we divide $a$, $b$, $c$ and $d$ by $ad-bc$?
Once again, this truly seems like magic. It's like ''hey, swap these fellers around, twiddle these signs a little, and poof, you're inverse!''.
Thanks in advance for any help :)

You can derive it by simultaneous transforming $(A|I)\to(I|A^{-1})$. If $a \ne 0$ and $ad-bc\ne 0$ we get: 
$$
\left[
\begin{array}{rr|rr}
a  & b & 1 & 0 \\
c & d & 0 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & b/a & 1/a & 0 \\
c & d & 0 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & b/a & 1/a & 0 \\
0 & d - cb/a & -c/a & 1
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & b/a & 1/a & 0 \\
0 & (ad - bc)/a & -c/a & 1
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & b/a & 1/a & 0 \\
0 & 1 & -c/(ad-bc) & a/(ad-bc)
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & 0 & 1/a + bc/a(ad-bc) & -b/(ad-bc) \\
0 & 1 & -c/(ad-bc) & a/(ad-bc)
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & 0 & ((ad-bc)+bc)/a(ad-bc) & -b/(ad-bc) \\
0 & 1 & -c/(ad-bc) & a/(ad-bc)
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1  & 0 & d/(ad-bc) & -b/(ad-bc) \\
0 & 1 & -c/(ad-bc) & a/(ad-bc)
\end{array}
\right]
$$
If $a = 0$ and $ad - bc \ne 0$ we have $b\ne 0$ and $c\ne 0$ and it goes like this:
$$
\left[
\begin{array}{rr|rr}
0  & b & 1 & 0 \\
c & d & 0 & 1
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
c & d & 0 & 1 \\
0 & b & 1 & 0
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1 & d/c & 0 & 1/c \\
0 & 1 & 1/b & 0
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1 & 0 & -d/cb & 1/c \\
0 & 1 & 1/b & 0
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1 & 0 & d/(-bc) & -b/(-bc) \\
0 & 1 & -c/(-bc) & 0/(-bc)
\end{array}
\right]
\to
\left[
\begin{array}{rr|rr}
1 & 0 & d/(ad-bc) & -b/(ad-bc) \\
0 & 1 & -c/(ad-bc) & 0/(ad-bc)
\end{array}
\right]
$$
