$8 . F i n d t h e i n v e r s e o f e a c h o f t h e f o l l o w i n g m a t r i c e s u \sin g e l e m e n t a r y c o l u m n t r a n s f o r m a t i o n s : i) [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}]$

A = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] A = A I [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] C_{2} \to \frac{1}{2} C_{1} + C_{2} [\begin{matrix} 2 & 0 & 4 \\ 4 & 2 & 2 \\ 3 & - \frac{1}{2} & 7 \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} 1 & \frac{1}{2} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] C_{1} \to C_{1} - 2 C_{3} [\begin{matrix} - 6 & 0 & 4 \\ 0 & 2 & 2 \\ - 11 & - \frac{1}{2} & 7 \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} 1 & \frac{1}{2} & 0 \\ 0 & 1 & 0 \\ - 2 & 0 & 1 \end{matrix}] C_{3} \to C_{3} - C_{2} [\begin{matrix} - 6 & 0 & 4 \\ 0 & 2 & 0 \\ - 11 & - \frac{1}{2} & \frac{15}{2} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} 1 & \frac{1}{2} & - \frac{1}{2} \\ 0 & 1 & - 1 \\ - 2 & 0 & 1 \end{matrix}] C_{1} \to - \frac{4}{6} C_{1} [\begin{matrix} 4 & 0 & 4 \\ 0 & 2 & 0 \\ \frac{22}{3} & - \frac{1}{2} & \frac{15}{2} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - \frac{2}{3} & \frac{1}{2} & - \frac{1}{2} \\ 0 & 1 & - 1 \\ \frac{4}{3} & 0 & 1 \end{matrix}] C_{3} \to C_{3} - C_{1} [\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ \frac{22}{3} & - \frac{1}{2} & \frac{15}{2} - \frac{22}{3} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - \frac{2}{3} & \frac{1}{2} & \frac{1}{6} \\ 0 & 1 & - 1 \\ \frac{4}{3} & 0 & - \frac{1}{3} \end{matrix}] [\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ \frac{22}{3} & - \frac{1}{2} & \frac{1}{6} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - \frac{2}{3} & \frac{1}{2} & \frac{1}{6} \\ 0 & 1 & - 1 \\ \frac{4}{3} & 0 & - \frac{1}{3} \end{matrix}] C_{2} \to C_{2} + 3 C_{3} [\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ \frac{22}{3} & 0 & \frac{1}{6} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - \frac{2}{3} & 1 & \frac{1}{6} \\ 0 & - 2 & - 1 \\ \frac{4}{3} & - 1 & - \frac{1}{3} \end{matrix}] C_{1} \to C_{1} - 44 C_{3} [\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & \frac{1}{6} \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - 8 & 1 & \frac{1}{6} \\ 44 & - 2 & - 1 \\ 16 & - 1 & - \frac{1}{3} \end{matrix}] C_{1} \to \frac{C_{1}}{4} C_{2} \to \frac{C_{2}}{2} C_{3} \to 6 C_{3} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}] [\begin{matrix} - 2 & \frac{1}{2} & 1 \\ 11 & - 1 & - 6 \\ 4 & - \frac{1}{2} & - 2 \end{matrix}] A^{- 1} = [\begin{matrix} - 2 & \frac{1}{2} & 1 \\ 11 & - 1 & - 6 \\ 4 & - \frac{1}{2} & - 2 \end{matrix}]

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8 . F i n d t h e i n v e r s e o f e a c h o f t h e f o l l o w i n g m a t r i c e s u sin g e l e m e n t a r y c o l u m n t r a n s f o r m a t i o n s : i ) 2 - 1 4 4 0 2 3 - 2 7

$8 . F i n d t h e i n v e r s e o f e a c h o f t h e f o l l o w i n g m a t r i c e s u \sin g e l e m e n t a r y c o l u m n t r a n s f o r m a t i o n s : i) [\begin{matrix} 2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7 \end{matrix}]$