Chapter 2.1, Problem 13E

Prove the following facts:
a. The $2\times 2$ matrix
$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
is invertible if and only if $ad-bc\ne 0$ . Hint: Consider the cases a $a\ne 0$ and $a=0$ separately.
b.I
$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
is invertible, then
${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$
[The formula in part (b) is worth memorizing.]