Find the Inverse [[1,2,3],[2,5,3],[1,0,8]]

Problem

[[1,2,3],[2,5,3],[1,0,8]]

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],[2,5,3,|,0,1,0],[1,0,8,|,0,0,1]]

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

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

3. Eliminate the entry in the second column below the second pivot by performing (R_3)←(R_3)+2*(R_2)

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

4. Normalize the third row by multiplying by −1 performing (R_3)←−(R_3)

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

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

[[1,2,0,|,−14,6,3],[0,1,0,|,13,−5,−3],[0,0,1,|,5,−2,−1]]

6. Eliminate the entry in the second column above the first pivot by performing (R_1)←(R_1)−2*(R_2)

[[1,0,0,|,−40,16,9],[0,1,0,|,13,−5,−3],[0,0,1,|,5,−2,−1]]

Final Answer

[[1,2,3],[2,5,3],[1,0,8]](−1)=[[−40,16,9],[13,−5,−3],[5,−2,−1]]

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