Guided Example #1 - Inverse

Interchange leading diagonal elements:
${\displaystyle 4\to t;3}$; ${\displaystyle 3\to ;4}$
${\displaystyle \left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]}$

Change signs of the other 2 elements:
${\displaystyle \left[\begin{array}{cc}3& -2\\ -5& 4\end{array}\right]}$

Find the determinant of A
${\displaystyle \left|\begin{array}{cc}4& 2\\ 5& 3\end{array}\right|=4\left(3\right)-5\left(2\right)=12-10=2}$

Multiply result of [2] by ${\displaystyle \frac{1}{detA}}$
${\displaystyle A^{-1}=\frac{1}{detA}[(3,-2),(-54)]=\frac{1}{2}\left[\begin{array}{cc}3& -2\\ -5& 4\end{array}\right]=\left[\begin{array}{cc}3⁄2& -1\\ -5⁄2& 2\end{array}\right]}$.

Interchange leading diagonal elements:
${\displaystyle 4\to t;3}$; ${\displaystyle 3\to ;4}$
${\displaystyle \left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]}$

Change signs of the other 2 elements:
${\displaystyle \left[\begin{array}{cc}3& -2\\ -5& 4\end{array}\right]}$

Find the determinant of A
${\displaystyle \left|\begin{array}{cc}4& 2\\ 5& 3\end{array}\right|=4\left(3\right)-5\left(2\right)=12-10=2}$

Multiply result of [2] by ${\displaystyle \frac{1}{detA}}$
${\displaystyle A^{-1}=\frac{1}{detA}[(3,-2),(-54)]=\frac{1}{2}\left[\begin{array}{cc}3& -2\\ -5& 4\end{array}\right]=\left[\begin{array}{cc}3⁄2& -1\\ -5⁄2& 2\end{array}\right]}$.