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

Problem

[[1,3,−1],[−3,−5,0],[4,5,1]]

Solution

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

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

2. Eliminate the entries in the first column below the first pivot by adding 3 times the first row to the second row, and subtracting 4 times the first row from the third row.

[[1,3,−1,|,1,0,0],[0,4,−3,|,3,1,0],[0,−7,5,|,−4,0,1]]

3. Normalize the second row by dividing it by 4 to create a pivot of 1.

[[1,3,−1,|,1,0,0],[0,1,−0.75,|,0.75,0.25,0],[0,−7,5,|,−4,0,1]]

4. Eliminate the entries in the second column above and below the second pivot by subtracting 3 times the second row from the first row, and adding 7 times the second row to the third row.

[[1,0,1.25,|,−1.25,−0.75,0],[0,1,−0.75,|,0.75,0.25,0],[0,0,−0.25,|,1.25,1.75,1]]

5. Normalize the third row by multiplying it by -4 to create a pivot of 1.

[[1,0,1.25,|,−1.25,−0.75,0],[0,1,−0.75,|,0.75,0.25,0],[0,0,1,|,−5,−7,−4]]

6. Eliminate the entries in the third column above the third pivot by subtracting 1.25 times the third row from the first row, and adding 0.75 times the third row to the second row.

[[1,0,0,|,5,8,5],[0,1,0,|,−3,−5,−3],[0,0,1,|,−5,−7,−4]]

Final Answer

[[1,3,−1],[−3,−5,0],[4,5,1]](−1)=[[5,8,5],[−3,−5,−3],[−5,−7,−4]]

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