1) Dada la matriz A \( A=\left(\begin{array}{ccc}3 & 1 & 4 \\ 6 & -2 & 5 \\ 2 & -1 & -3\end{array}\right) \) Hallar lo inversa por Gavss-Jordan

Find the inverse matrix of \( \begin{pmatrix}3 & 1 & 4 \\ 6 & -2 & 5 \\ 2 & -1 & -3\end{pmatrix} \). Matrices by following steps: - step0: Find the matrix inverse: \(\left[\begin{array}{rrr}{3}&{1}&{4}\\{6}&{-2}&{5}\\{2}&{-1}&{-3}\end{array}\right]\) - step1: Begin by adjoining the identity matrix to form the matrix: \(\left[\begin{array}{rrr|rrr}{3}&{1}&{4}&{1}&{0}&{0}\\{6}&{-2}&{5}&{0}&{1}&{0}\\{2}&{-1}&{-3}&{0}&{0}&{1}\end{array}\right]\) - step2: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{6}&{-2}&{5}&{0}&{1}&{0}\\{2}&{-1}&{-3}&{0}&{0}&{1}\end{array}\right]\) - step3: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{-4}&{-3}&{-2}&{1}&{0}\\{2}&{-1}&{-3}&{0}&{0}&{1}\end{array}\right]\) - step4: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{-4}&{-3}&{-2}&{1}&{0}\\{0}&{-\frac{5}{3}}&{-\frac{17}{3}}&{-\frac{2}{3}}&{0}&{1}\end{array}\right]\) - step5: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{1}&{\frac{3}{4}}&{\frac{1}{2}}&{-\frac{1}{4}}&{0}\\{0}&{-\frac{5}{3}}&{-\frac{17}{3}}&{-\frac{2}{3}}&{0}&{1}\end{array}\right]\) - step6: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{1}&{\frac{3}{4}}&{\frac{1}{2}}&{-\frac{1}{4}}&{0}\\{0}&{0}&{-\frac{53}{12}}&{\frac{1}{6}}&{-\frac{5}{12}}&{1}\end{array}\right]\) - step7: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{1}&{\frac{3}{4}}&{\frac{1}{2}}&{-\frac{1}{4}}&{0}\\{0}&{0}&{1}&{-\frac{2}{53}}&{\frac{5}{53}}&{-\frac{12}{53}}\end{array}\right]\) - step8: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{\frac{4}{3}}&{\frac{1}{3}}&{0}&{0}\\{0}&{1}&{0}&{\frac{28}{53}}&{-\frac{17}{53}}&{\frac{9}{53}}\\{0}&{0}&{1}&{-\frac{2}{53}}&{\frac{5}{53}}&{-\frac{12}{53}}\end{array}\right]\) - step9: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{\frac{1}{3}}&{0}&{\frac{61}{159}}&{-\frac{20}{159}}&{\frac{16}{53}}\\{0}&{1}&{0}&{\frac{28}{53}}&{-\frac{17}{53}}&{\frac{9}{53}}\\{0}&{0}&{1}&{-\frac{2}{53}}&{\frac{5}{53}}&{-\frac{12}{53}}\end{array}\right]\) - step10: Simplify the row: \(\left[\begin{array}{rrr|rrr}{1}&{0}&{0}&{\frac{11}{53}}&{-\frac{1}{53}}&{\frac{13}{53}}\\{0}&{1}&{0}&{\frac{28}{53}}&{-\frac{17}{53}}&{\frac{9}{53}}\\{0}&{0}&{1}&{-\frac{2}{53}}&{\frac{5}{53}}&{-\frac{12}{53}}\end{array}\right]\) - step11: Write the right half matrix: \(\left[\begin{array}{rrr}{\frac{11}{53}}&{-\frac{1}{53}}&{\frac{13}{53}}\\{\frac{28}{53}}&{-\frac{17}{53}}&{\frac{9}{53}}\\{-\frac{2}{53}}&{\frac{5}{53}}&{-\frac{12}{53}}\end{array}\right]\) La matriz inversa de la matriz \( A \) es: \[ A^{-1} = \left[\begin{array}{rrr}\frac{11}{53} & -\frac{1}{53} & \frac{13}{53} \\ \frac{28}{53} & -\frac{17}{53} & \frac{9}{53} \\ -\frac{2}{53} & \frac{5}{53} & -\frac{12}{53}\end{array}\right] \]