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

Problem

[[1,0,−2],[−3,1,4],[2,−3,4]]

Solution

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

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

2. Eliminate the entries in the first column below the pivot by performing row operations (R_2)←(R_2)+3*(R_1) and (R_3)←(R_3)−2*(R_1)

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

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

[[1,0,−2,|,1,0,0],[0,1,−2,|,3,1,0],[0,0,2,|,7,3,1]]

4. Normalize the third row to create a pivot of 1 by performing the row operation (R_3)←1/2*(R_3)

[[1,0,−2,|,1,0,0],[0,1,−2,|,3,1,0],[0,0,1,|,3.5,1.5,0.5]]

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

[[1,0,0,|,8,3,1],[0,1,0,|,10,4,1],[0,0,1,|,3.5,1.5,0.5]]

6. Identify the resulting matrix on the right side as the inverse A(−1)

Final Answer

[[1,0,−2],[−3,1,4],[2,−3,4]](−1)=[[8,3,1],[10,4,1],[3.5,1.5,0.5]]

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