Question

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

Answer 1

Given,

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

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

$R_2\to R_2-2R_1$

$\left[\begin{array}{cc}1& 2\\ 0& -5\end{array}|\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]$

$\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}|\begin{array}{cc}1& 0\\ \frac{2}{5}& \frac{-1}{5}\end{array}\right]$

$R_1+R_1-2R_2$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}|\begin{array}{cc}\frac{1}{5}& \frac{2}{5}\\ \frac{2}{5}& \frac{-1}{5}\end{array}\right]$

Hence, $A^{-1}=\left[\begin{array}{cc}\frac{1}{5}& \frac{2}{5}\\ \frac{2}{5}& \frac{-1}{5}\end{array}\right]$