Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find $A^{-1}$ by elementary transformation.

Answer 1

Given, $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

Noe, $A+IA$

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$

$R_2\to R_2-3R_1$

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

$R_1\to R_1+R_2$

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

$R_2\to R_2/2$

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

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