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

Question:

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

$\left[\begin{array}{ll}4 & 5 \\ 3 & 4\end{array}\right]$

Solution:

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

We know that A = IA

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

$\Rightarrow\left[\begin{array}{ll}1 & 1 \\ 3 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]=\left[\begin{array}{lr}1 & -1 \\ -3 & 4\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-3 \mathrm{R}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{cr}4 & -5 \\ -3 & 4\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\right)$

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