Find the Inverse [[0,1,2],[1,0,3],[4,-3,8]]

Problem

[[0,1,2],[1,0,3],[4,−3,8]]

Solution

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

[[0,1,2,|,1,0,0],[1,0,3,|,0,1,0],[4,−3,8,|,0,0,1]]

2. Swap the first and second rows to get a leading 1 in the first column.

[[1,0,3,|,0,1,0],[0,1,2,|,1,0,0],[4,−3,8,|,0,0,1]]

3. Eliminate the 4 in the third row by performing the row operation (R_3)−4*(R_1)⇒(R_3)

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

4. Eliminate the −3 in the third row by performing the row operation (R_3)+3*(R_2)⇒(R_3)

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

5. Normalize the third row by dividing by 2 ((R_3)/2⇒(R_3).

[[1,0,3,|,0,1,0],[0,1,2,|,1,0,0],[0,0,1,|,1.5,−2,0.5]]

6. Eliminate the 2 in the second row by performing (R_2)−2*(R_3)⇒(R_2)

[[1,0,3,|,0,1,0],[0,1,0,|,−2,4,−1],[0,0,1,|,1.5,−2,0.5]]

7. Eliminate the 3 in the first row by performing (R_1)−3*(R_3)⇒(R_1)

[[1,0,0,|,−4.5,7,−1.5],[0,1,0,|,−2,4,−1],[0,0,1,|,1.5,−2,0.5]]

Final Answer

[[0,1,2],[1,0,3],[4,−3,8]](−1)=[[−4.5,7,−1.5],[−2,4,−1],[1.5,−2,0.5]]

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