find the inverse of the matrix 1 3 -2 using elementart transformation ? -3 0 -5 2 5 0

let the given matrix is $A=\left[\begin{array}{c}1\\ -3\\ 2\end{array}\begin{array}{c}3\\ 0\\ 5\end{array}\begin{array}{c}-2\\ -5\\ 0\end{array}\right]$
let us find the inverse of matrix A by elementary row operation.
$\left[\overline{)\begin{array}{c}1\\ -3\\ 2\end{array}\begin{array}{c}3\\ 0\\ 5\end{array}\begin{array}{c}-2\\ -5\\ 0\end{array}}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]$ interchange $R_2 and R_3$

$\left[\overline{)\begin{array}{c}1\\ 2\\ -3\end{array}\begin{array}{c}3\\ 5\\ 0\end{array}\begin{array}{c}-2\\ 0\\ -5\end{array}}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ apply $R_2\to R_2/5 and R_3\to R_3/(-5)$

$\left[\overline{)\begin{array}{c}1\\ 2/5\\ 3/5\end{array}\begin{array}{c}3\\ 1\\ 0\end{array}\begin{array}{c}-2\\ 0\\ 1\end{array}}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ -1/5\end{array}\begin{array}{c}0\\ 1/5\\ 0\end{array}\right]$ apply $R_1\to 2R_3+R_1$

$\left[\overline{)\begin{array}{c}11/5\\ 2/5\\ 3/5\end{array}\begin{array}{c}3\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}-2/5\\ 0\\ -1/5\end{array}\begin{array}{c}0\\ 1/5\\ 0\end{array}\right]$ apply $R_1\to R_1-3R_2$

$\left[\overline{)\begin{array}{c}1\\ 2/5\\ 3/5\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}-2/5\\ 0\\ -1/5\end{array}\begin{array}{c}-3/5\\ 1/5\\ 0\end{array}\right]$ apply $R_3\to R_3-\frac{3}{5}{R}_1 and R_2\to R_2-\frac{2}{5}{R}_1$

$\left[\overline{)\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}}\begin{array}{c}1\\ -2/5\\ -3/5\end{array}\begin{array}{c}-2/5\\ 4/5\\ 1/25\end{array}\begin{array}{c}-3/5\\ 11/25\\ 9/25\end{array}\right]$
thus inverse of the given matrix is
$A^{-1}=\left[\begin{array}{c}1\\ -2/5\\ -3/5\end{array}\begin{array}{c}-2/5\\ 4/5\\ 1/25\end{array}\begin{array}{c}-3/5\\ 11/25\\ 9/25\end{array}\right]$

hope this helps you