Question

Using elementary tansormations, find the inverse of each of the matrices, if it exists in
$\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$

Answer 1

$A=\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$

$|A|=7-6=1$

Now, $A=IA$

$\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$

$\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]A$ $(R_2\to R_2-2R_1)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]A$ $(R_1\to R_1-3R_2)$

Hence, $A^{-1}=\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$