Find the Inverse [[1,0,-4],[-3,3,1],[5,5,2]]

Problem

[[1,0,−4],[−3,3,1],[5,5,2]]

Solution

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

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

2. Eliminate the entries in the first column below the pivot by adding 3 times the first row to the second row, and subtracting 5 times the first row from the third row.

[[1,0,−4,|,1,0,0],[0,3,−11,|,3,1,0],[0,5,22,|,−5,0,1]]

3. Normalize the second row by dividing it by 3

[[1,0,−4,|,1,0,0],[0,1,−11/3,|,1,1/3,0],[0,5,22,|,−5,0,1]]

4. Eliminate the entry in the third row, second column by subtracting 5 times the second row from the third row.

[[1,0,−4,|,1,0,0],[0,1,−11/3,|,1,1/3,0],[0,0,121/3,|,−10,−5/3,1]]

5. Normalize the third row by multiplying it by 3/121

[[1,0,−4,|,1,0,0],[0,1,−11/3,|,1,1/3,0],[0,0,1,|,−30/121,−5/121,3/121]]

6. Eliminate the entries in the third column above the pivot by adding 4 times the third row to the first row, and adding 11/3 times the third row to the second row.

[[1,0,0,|,1/121,−20/121,12/121],[0,1,0,|,11/121,22/121,11/121],[0,0,1,|,−30/121,−5/121,3/121]]

7. Simplify the fractions in the resulting inverse matrix.

[[1/121,−20/121,12/121],[1/11,2/11,1/11],[−30/121,−5/121,3/121]]

Final Answer

[[1,0,−4],[−3,3,1],[5,5,2]](−1)=[[1/121,−20/121,12/121],[1/11,2/11,1/11],[−30/121,−5/121,3/121]]

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