Question

Find the inverse of the matrices, if it exists.
$\left[\begin{array}{cc}4& 5\\ 3& 4\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}4& 5\\ 3& 4\end{array}\right]$

We know that $A=IA$

$\therefore A=\left[\begin{array}{cc}4& 5\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$

$\Rightarrow \left[\begin{array}{cc}1& 1\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]A(R_1\to R_1-R_2)$

$\Rightarrow \left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& -1\\ -3& 4\end{array}\right]A(R_2\to R_2-{3R}_1)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}4& -5\\ -3& 4\end{array}\right]A(R_1\to R_1-R_2)$

$\therefore A^{-1}=\left[\begin{array}{cc}4& -5\\ -3& 4\end{array}\right]$