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

Problem

[[1,1,0],[3,1,1],[0,1,−1]](−1)

Solution

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

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

2. Eliminate the entry in the second row, first column by performing the row operation (R_2)−3*(R_1)⇒(R_2)

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

3. Swap the second and third rows ((R_2)⇔(R_3) to make the pivot calculation easier.

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

4. Eliminate the entry in the third row, second column by performing (R_3)+2*(R_2)⇒(R_3)

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

5. Normalize the third row by multiplying by −1 (−1*(R_3)⇒(R_3).

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

6. Eliminate the entry in the second row, third column by performing (R_2)+(R_3)⇒(R_2)

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

7. Eliminate the entry in the first row, second column by performing (R_1)−(R_2)⇒(R_1)

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

Final Answer

[[1,1,0],[3,1,1],[0,1,−1]](−1)=[[−2,1,1],[3,−1,−1],[3,−1,−2]]

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