Find the inverse of each of the following matrices by using elementary row transformations:
Question:

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ll}5 & 2 \\ 2 & 1\end{array}\right]$

Solution:

$A=\left[\begin{array}{ll}5 & 2\end{array}\right.$

$\left.\begin{array}{ll}2 & 1\end{array}\right]$

We know

$A=I A$

$\Rightarrow\left[\begin{array}{ll}5 & 2\end{array}\right.$

$2 \quad 1]=\left[\begin{array}{ll}1 & 0\end{array}\right.$

$0 \quad 1] A \quad\left[\right.$ Applying $\left.\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-2 \mathrm{R}_{2}\right]$

$\Rightarrow\left[\begin{array}{ll}1 & 0\end{array}\right.$

$2 \quad 1]=\left[\begin{array}{ll}1 & -2\end{array}\right.$

$0 \quad 1] A$

$\Rightarrow\left[\begin{array}{ll}1 & 0\end{array}\right.$

$2-2 \quad 1]=\left[\begin{array}{ll}1 & -2\end{array}\right.$

$0-2 \quad 1+4] A \quad$ [Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}$ ]

$\Rightarrow\left[\begin{array}{ll}1 & 0\end{array}\right.$

$\left.\begin{array}{ll}0 & 1\end{array}\right]=\left[\begin{array}{ll}1 & -2\end{array}\right.$

$\Rightarrow A^{-1}=\left[\begin{array}{ll}1 & -2\end{array}\right.$

$\left.\begin{array}{ll}-2 & 5\end{array}\right]$