Question

Using elementary tansormations, find the inverse of each of the matrices, if it exists in
$\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

Answer 1

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

$|A|=2-1=1\ne 0$

Hence, $A^{-1}$ exists.

Now, $A=IA$

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

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

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

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