Question

Find the inverse of the matrices, if it exists.
$\left[\begin{array}{cc}3& -1\\ -4& 2\end{array}\right]$

Answer 1

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

We know that $A=IA$

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

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

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

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=A\left[\begin{array}{cc}1& \frac{1}{2}\\ 2& \frac{3}{2}\end{array}\right](C_1\to \frac{1}{2}{C}_2)$

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