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

Problem

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

2. Eliminate the entry in the first column of the second row by adding the first row to the second row ((R_2)←(R_2)+(R_1).

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

3. Create a leading one in the second row by multiplying the second row by −1/3 ((R_2)←−1/3*(R_2).

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

4. Eliminate the entries in the second column for the first and third rows ((R_1)←(R_1)+(R_2) and (R_3)←(R_3)+2*(R_2).

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

5. Create a leading one in the third row by multiplying the third row by 3 ((R_3)←3*(R_3).

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

6. Eliminate the entries in the third column for the first and second rows ((R_1)←(R_1)+1/3*(R_3) and (R_2)←(R_2)+1/3*(R_3).

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

Final Answer

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

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