Question

Find the inverse of the matrix $A=\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right]$ using elementry transformations.

Answer 1

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

$\therefore A=IA$

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

Applying $R_2\to R_2-R_1$

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

Applying $R_1\to R_1-2R_2$

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

We know $A\cdot A^{-1}=I$

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