Question

Find the inverse of each of the matrices , if it exists.
$\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$
We know that $A=IA$
$\therefore \left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]A(R_1\to R_1-R_2)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]A(R_2\to R_2-R_1)$

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