Question

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

$\left[\begin{array}{cc}3& 10\\ 2& 7\end{array}\right]$

Answer 1

$$\begin{gathered} A=\left[310 \\ 27\right] \\ \mathrm{We}\mathrm{know} \\ A=IA \\ \Rightarrow \left[310 \\ 27\right]=\left[10 \\ 01\right]A \\ \Rightarrow \left[3-210-7 \\ 27\right]=\left[1-00-1 \\ 01\right]A[\mathrm{Applying}{R}_1\to R_1-R_2] \\ \Rightarrow \left[13 \\ 27\right]=\left[1-1 \\ 01\right]A \\ \Rightarrow \left[13 \\ 2-27-6\right]=\left[1-1 \\ 0-21+2\right][\mathrm{Applying}{R}_2\to R_2-2R_1] \\ \Rightarrow \left[13 \\ 01\right]=\left[1-1 \\ -23\right]A \\ \Rightarrow \left[13-3 \\ 01\right]=\left[1+6-1-9 \\ -23\right]A[\mathrm{Applying}{R}_1\to R_1-3R_2] \\ \Rightarrow \left[10 \\ 01\right]=\left[7-10 \\ -23\right]A \\ \Rightarrow A^{-1}=\left[7-10 \\ -23\right] \end{gathered}$$