Question

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

$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]$

Answer 1

$$\begin{gathered} A=\left[25 \\ 13\right] \\ \mathrm{We}\mathrm{know} \\ A=IA \\ \Rightarrow \left[25 \\ 13\right]=\left[10 \\ 01\right]A \\ \Rightarrow \left[2-15-3 \\ 13\right]=\left[1-00-1 \\ 01\right]A[\mathrm{Applying}{R}_1\to R_1-R_2] \\ \Rightarrow \left[12 \\ 13\right]=\left[1-1 \\ 01\right]A \\ \Rightarrow \left[12 \\ 1-13-2\right]=\left[1-1 \\ 0-11+1\right]A[\mathrm{Applying}{R}_2\to R_2-R_1] \\ \Rightarrow \left[12 \\ 01\right]=\left[1-1 \\ -12\right]A \\ \Rightarrow \left[10 \\ 01\right]=\left[1+2-1-4 \\ -12\right]A[\mathrm{Applying}{R}_1\to R_1-2R_2] \\ \Rightarrow \left[10 \\ 01\right]=\left[3-5 \\ -12\right]A \\ \Rightarrow A^{-1}=\left[3-5 \\ -12\right] \end{gathered}$$