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

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

$\left[\begin{array}{lr}3 & -1 \\ -4 & 2\end{array}\right]$

Solution:

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

We know that $A=A l$

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

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

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right]=A\left[\begin{array}{ll}1 & 1 \\ 2 & 3\end{array}\right] \quad\left(\mathrm{C}_{2} \rightarrow \mathrm{C}_{2}+\mathrm{C}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=A\left[\begin{array}{ll}1 & \frac{1}{2} \\ 2 & \frac{3}{2}\end{array}\right] \quad\left(\mathrm{C}_{2} \rightarrow \frac{1}{2} \mathrm{C}_{2}\right)$

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