Find the Inverse [[1,2,3],[0,1,-1],[2,2,2]]

Problem

[[1,2,3],[0,1,−1],[2,2,2]](−1)

Solution

1. Set up the augmented matrix by placing the identity matrix I to the right of the given matrix A

[[1,2,3,|,1,0,0],[0,1,−1,|,0,1,0],[2,2,2,|,0,0,1]]

2. Eliminate the first element of the third row by performing the row operation (R_3)→(R_3)−2*(R_1)

[[1,2,3,|,1,0,0],[0,1,−1,|,0,1,0],[0,−2,−4,|,−2,0,1]]

3. Eliminate the second element of the third row by performing the row operation (R_3)→(R_3)+2*(R_2)

[[1,2,3,|,1,0,0],[0,1,−1,|,0,1,0],[0,0,−6,|,−2,2,1]]

4. Normalize the third row by dividing by −6 using the operation (R_3)→−1/6*(R_3)

[[1,2,3,|,1,0,0],[0,1,−1,|,0,1,0],[0,0,1,|,1/3,−1/3,−1/6]]

5. Eliminate the elements above the pivot in the third column using (R_2)→(R_2)+(R_3) and (R_1)→(R_1)−3*(R_3)

[[1,2,0,|,0,1,1/2],[0,1,0,|,1/3,2/3,−1/6],[0,0,1,|,1/3,−1/3,−1/6]]

6. Eliminate the element above the pivot in the second column using (R_1)→(R_1)−2*(R_2)

[[1,0,0,|,−2/3,−1/3,5/6],[0,1,0,|,1/3,2/3,−1/6],[0,0,1,|,1/3,−1/3,−1/6]]

Final Answer

[[1,2,3],[0,1,−1],[2,2,2]](−1)=[[−2/3,−1/3,5/6],[1/3,2/3,−1/6],[1/3,−1/3,−1/6]]

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