# Find the inverse of each of the matrices, if it exists.

Question:

Find the inverse of each of the matrices, if it exists.

$\left[\begin{array}{ll}2 & 1 \\ 7 & 4\end{array}\right]$

Solution:

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

We know that $A=I A$

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

$\Rightarrow\left[\begin{array}{ll}1 & \frac{1}{2} \\ 7 & 4\end{array}\right]=\left[\begin{array}{ll}\frac{1}{2} & 0 \\ 0 & 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \frac{1}{2} \mathrm{R}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & \frac{1}{2} \\ 0 & \frac{1}{2}\end{array}\right]=\left[\begin{array}{ll}\frac{1}{2} & 0 \\ -\frac{7}{2} & 1\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-7 \mathrm{R}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & \frac{1}{2}\end{array}\right]=\left[\begin{array}{lr}4 & -1 \\ -\frac{7}{2} & 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right]=\left[\begin{array}{lr}4 & -1 \\ -7 & 2\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow 2 \mathrm{R}_{2}\right)$

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