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

Problem

[[1,0,0],[2,4,0],[1,−1,1]](−1)

Solution

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

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

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

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

3. Normalize the second row by dividing (R_2) by 4 to create a pivot of 1

[[1,0,0,|,1,0,0],[0,1,0,|,−0.5,0.25,0],[0,−1,1,|,−1,0,1]]

4. Eliminate the entry below the second pivot by performing the row operation (R_3)←(R_3)+(R_2)

[[1,0,0,|,1,0,0],[0,1,0,|,−0.5,0.25,0],[0,0,1,|,−1.5,0.25,1]]

5. Identify the inverse matrix from the right side of the augmented matrix now that the left side is the identity matrix.

[[1,0,0],[−0.5,0.25,0],[−1.5,0.25,1]]

Final Answer

[[1,0,0],[2,4,0],[1,−1,1]](−1)=[[1,0,0],[−1/2,1/4,0],[−3/2,1/4,1]]

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