Question

By using elementary transformations, find the inverse of the matrix.
$A=\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$.

Answer 1

Given:
$A=\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$
Using Row operations
$A=IA$
$\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$
$R_2^1=R_2-R_1$
$\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]A$
$R_1^1=R_1-3R_2$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]A$
$\therefore I=A^{-1}A$
$\therefore A^{-1}=\left[\begin{array}{cc}7& -3\\ -2& 1\end{array}\right]$.