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$Q . 2 B y e l e m e n t o r y r o w t r a n s f o r m a t i o n, f i n d t h e i n v e r s e o f A = [\begin{matrix} 3 & - 1 & 2 \\ - 4 & 0 & 5 \\ 1 & 3 & 2 \end{matrix}]$

A = [\begin{matrix} 3 & - 1 & 2 \\ - 4 & 0 & 5 \\ 1 & 3 & 2 \end{matrix}] [\begin{matrix} 3 & - 1 & 2 \\ - 4 & 0 & 5 \\ 1 & 3 & 2 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A R_{1} \leftrightarrow R_{3} [\begin{matrix} 1 & 3 & 2 \\ - 4 & 0 & 5 \\ 3 & - 1 & 2 \end{matrix}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}] A R_{2} \to R_{2} + 4 R_{1} R_{3} \to R_{3} - 3 R_{1} [\begin{matrix} 1 & 3 & 2 \\ 0 & 12 & 13 \\ 0 & - 10 & - 4 \end{matrix}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 4 \\ 1 & 0 & - 3 \end{matrix}] A R_{2} \to \frac{R_{2}}{12} [\begin{matrix} 1 & 3 & 2 \\ 0 & 1 & \frac{13}{12} \\ 0 & - 10 & - 4 \end{matrix}] = [\begin{matrix} 0 & 0 & 1 \\ 0 & \frac{1}{12} & \frac{1}{3} \\ 1 & 0 & - 3 \end{matrix}] A R_{1} \to R_{1} - 3 R_{2} R_{3} \to R_{3} + 10 R_{2} [\begin{matrix} 1 & 0 & - \frac{5}{4} \\ 0 & 1 & \frac{13}{12} \\ 0 & 0 & \frac{41}{6} \end{matrix}] = [\begin{matrix} 0 & - \frac{1}{4} & 0 \\ 0 & \frac{1}{12} & \frac{1}{3} \\ 1 & \frac{5}{6} & \frac{1}{3} \end{matrix}] A R_{3} \to \frac{6}{41} R_{3} [\begin{matrix} 1 & 0 & - \frac{5}{4} \\ 0 & 1 & \frac{13}{12} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 0 & - \frac{1}{4} & 0 \\ 0 & \frac{1}{12} & \frac{1}{3} \\ \frac{6}{41} & \frac{5}{41} & \frac{2}{41} \end{matrix}] A R_{1} \to R_{1} + \frac{5}{4} R_{3} R_{2} \to R_{2} - \frac{13}{12} R_{3} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} \frac{30}{164} & - \frac{36}{164} & - \frac{10}{164} \\ - \frac{78}{492} & - \frac{24}{492} & \frac{138}{492} \\ \frac{6}{41} & \frac{5}{41} & \frac{2}{41} \end{matrix}] A [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} \frac{15}{82} & - \frac{4}{41} & - \frac{5}{82} \\ - \frac{13}{82} & - \frac{2}{41} & \frac{23}{82} \\ \frac{6}{41} & \frac{5}{41} & \frac{2}{41} \end{matrix}] A A^{- 1} = [\begin{matrix} \frac{15}{82} & - \frac{4}{41} & - \frac{5}{82} \\ - \frac{13}{82} & - \frac{2}{41} & \frac{23}{82} \\ \frac{6}{41} & \frac{5}{41} & \frac{2}{41} \end{matrix}]

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Solve this : Q . 2 B y e l e m e n t o r y r o w t r a n s f o r m a t i o n , f i n d t h e i n v e r s e o f A = 3 - 1 2 - 4 0 5 1 3 2

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$Q . 2 B y e l e m e n t o r y r o w t r a n s f o r m a t i o n, f i n d t h e i n v e r s e o f A = [\begin{matrix} 3 & - 1 & 2 \\ - 4 & 0 & 5 \\ 1 & 3 & 2 \end{matrix}]$