Find the Inverse [[1,-4,2],[-2,8,-9],[-1,7,0]]

Problem

[[1,−4,2],[−2,8,−9],[−1,7,0]]

Solution

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

[[1,−4,2,|,1,0,0],[−2,8,−9,|,0,1,0],[−1,7,0,|,0,0,1]]

2. Eliminate the entries in the first column below the first pivot by performing row operations (R_2)←(R_2)+2*(R_1) and (R_3)←(R_3)+(R_1)

[[1,−4,2,|,1,0,0],[0,0,−5,|,2,1,0],[0,3,2,|,1,0,1]]

3. Swap (R_2) and (R_3) to bring a non-zero value into the second pivot position.

[[1,−4,2,|,1,0,0],[0,3,2,|,1,0,1],[0,0,−5,|,2,1,0]]

4. Normalize the second and third rows by dividing (R_2) by 3 and (R_3) by −5

[[1,−4,2,|,1,0,0],[0,1,2/3,|,1/3,0,1/3],[0,0,1,|,−2/5,−1/5,0]]

5. Eliminate the entries above the third pivot by performing (R_2)←(R_2)−2/3*(R_3) and (R_1)←(R_1)−2*(R_3)

[[1,−4,0,|,9/5,2/5,0],[0,1,0,|,3/5,2/15,1/3],[0,0,1,|,−2/5,−1/5,0]]

6. Eliminate the entry above the second pivot by performing (R_1)←(R_1)+4*(R_2)

[[1,0,0,|,21/5,14/15,4/3],[0,1,0,|,3/5,2/15,1/3],[0,0,1,|,−2/5,−1/5,0]]

Final Answer

[[1,−4,2],[−2,8,−9],[−1,7,0]](−1)=[[21/5,14/15,4/3],[3/5,2/15,1/3],[−2/5,−1/5,0]]

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