Question

Find the inverse of the matrices, if it exists.
$\left[\begin{array}{cc}2& -6\\ 1& -2\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& -6\\ 1& -2\end{array}\right]$

We know that

$A=IA$

$A=\left[\begin{array}{cc}2& -6\\ 1& -2\end{array}\right]=A\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]=A\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right](C_2\to C_2+3C_1)$

$\Rightarrow \left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]=A\left[\begin{array}{cc}-2& 3\\ -1& 1\end{array}\right](C_1\to C_1-C_2)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=A\left[\begin{array}{cc}-1& 3\\ -\frac{1}{2}& 1\end{array}\right](C_1\to \frac{1}{2}{C}_1)$

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