Question

Find the inverse of the matrices, if it exists.
$\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$
We know that $A=IA$
$\therefore A=\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]=A\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

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

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

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=A\left[\begin{array}{cc}2& -1\\ -5& 3\end{array}\right](C_1\to C_1-C_2)$

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