Problem 11. Find the Inverse. 15 points. Find the inverse of the following matrix: \[ \left[\begin{array}{cccc} 1 & 1 & -1 & 1 \\ 2 & 1 & 1 & 0 \\ 2 & 1 & 0 & 1 \\ 2 & -1 & -1 & 3 \end{array}\right] \]

Step 1 ：Augment the given matrix with the 4x4 identity matrix: \[\left[\begin{array}{cccc|cccc} 1 & 1 & -1 & 1 & 1 & 0 & 0 & 0 \\ 2 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 2 & 1 & 0 & 1 & 0 & 0 & 1 & 0 \\ 2 & -1 & -1 & 3 & 0 & 0 & 0 & 1 \end{array}\right]\]

Step 2 ：Subtract 2 times the first row from the second, third and fourth rows: \[\left[\begin{array}{cccc|cccc} 1 & 1 & -1 & 1 & 1 & 0 & 0 & 0 \\ 0 & -1 & 3 & -2 & -2 & 1 & 0 & 0 \\ 0 & -1 & 2 & -1 & -2 & 0 & 1 & 0 \\ 0 & -3 & 1 & 1 & -2 & 0 & 0 & 1 \end{array}\right]\]

Step 3 ：Multiply the second row by -1 and add the second row to the third and three times the second row to the fourth: \[\left[\begin{array}{cccc|cccc} 1 & 1 & -1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & -3 & 2 & 2 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & -1 & 1 & 0 \\ 0 & 0 & -8 & -5 & -4 & -3 & 0 & 1 \end{array}\right]\]

Step 4 ：Multiply the third row by -1 and add 8 times the third row to the fourth: \[\left[\begin{array}{cccc|cccc} 1 & 1 & 0 & 1 & 1 & 1 & -1 & 0 \\ 0 & 1 & 0 & 2 & 2 & -1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & -5 & -4 & -5 & 8 & 1 \end{array}\right]\]

Step 5 ：Multiply the fourth row by -1/5 and add the first row to the second and the third row to the first: \[\left[\begin{array}{cccc|cccc} 1 & 0 & 0 & 0 & -3/5 & 0 & 3/5 & 1/5 \\ 0 & 1 & 0 & 0 & 2/5 & -2 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 1 & 4/5 & 1 & -8/5 & -1/5 \end{array}\right]\]

Step 6 ：The inverse of the given matrix is: \[\boxed{A^{-1} = \left[\begin{array}{cccc} -3/5 & 0 & 3/5 & 1/5 \\ 2/5 & -2 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 4/5 & 1 & -8/5 & -1/5 \end{array}\right]}\]