Question

Find the inverse of the following matrix using transformation method.
$\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

Answer 1

$A=\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$

$C = [ A|I]

$=\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$R_2\to R_2+2R_1$

$=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

$R_2\to R_2+R_1$

$=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$

$R_1\to R_1+R_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$

Hence, $A^{-1}=\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$