Question

Find the inverse of the matrices, if it exists.
$\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$

We know that $A=IA$

$\therefore A=\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]A$

$\Rightarrow \left[\begin{array}{cc}1& \frac{1}{2}\\ 7& 4\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& 1\end{array}\right]A(R_1\to \frac{1}{2}{R}_1)$

$\Rightarrow \left[\begin{array}{cc}1& \frac{1}{2}\\ 0& \frac{1}{2}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& 0\\ -\frac{7}{2}& 1\end{array}\right]A(R_2\to R_2-7R_1)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& \frac{1}{2}\end{array}\right]=\left[\begin{array}{cc}4& -1\\ -\frac{7}{2}& 1\end{array}\right]A(R_1\to R_1-R_2)$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}4& -1\\ -7& 2\end{array}\right]A(R_2\to 2R_2)$

$\therefore A^{-1}=\left[\begin{array}{cc}4& -1\\ -7& 2\end{array}\right]$