Question

Find the inverse of each of the matrices, if it exists.
$\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

Answer 1

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

We know that $A=IA$

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

$\Rightarrow \left[\begin{array}{cc}1& -1\\ 0& 5\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]A(R_2\to R_2-2R_1)$

$\Rightarrow \left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -\frac{2}{5}& \frac{1}{5}\end{array}\right]A(R_2\to \frac{1}{5}{R}_2)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\frac{3}{5}& \frac{1}{5}\\ -\frac{2}{5}& \frac{1}{5}\end{array}\right]A(R_1\to R_1+R_2)$

$\therefore A^{-1}=\left[\begin{array}{cc}\frac{3}{5}& \frac{1}{5}\\ -\frac{2}{5}& \frac{1}{5}\end{array}\right]$