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# Find the inverse of the following matrices if it exists using elementary operations $\begin{bmatrix}2 & 5\\1 & 3\end{bmatrix}$

Toolbox:
• In order to find the inverse using row elementary we write as A=IA.
$\begin{bmatrix}2 & 5\\1 & 3\end{bmatrix}=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}A$
Step 1: Apply $R_1\rightarrow R_1-R_2$
$\begin{bmatrix}1 & 2\\1 &3\end{bmatrix}=\begin{bmatrix}1 & -1\\0 & 1\end{bmatrix}A$
Step 2: Apply $R_2\rightarrow R_2-R_1$
$\begin{bmatrix}1 & 2\\0 &1\end{bmatrix}=\begin{bmatrix}3 & -1\\-1 & 2\end{bmatrix}A$
Step 3: Apply $R_1\rightarrow R_1-2R_2$
$\begin{bmatrix}1 & 0\\0 &1\end{bmatrix}=\begin{bmatrix}3 & -5\\-1 & 2\end{bmatrix}A$
Step 4: $A^{-1}=\begin{bmatrix}3 & -5\\-1 & 2\end{bmatrix}$