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

Problem

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

2. Eliminate the entry in the second row, third column by performing the row operation (R_2)→(R_2)+4*(R_3)

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

3. Eliminate the entry in the first row, third column by performing the row operation (R_1)→(R_1)+(R_3)

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

4. Eliminate the entry in the first row, second column by performing the row operation (R_1)→(R_1)−3*(R_2)

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

5. Identify the inverse matrix from the right side of the augmented matrix.

A(−1)=[[1,−3,−11],[0,1,4],[0,0,1]]

Final Answer

[[1,3,−1],[0,1,−4],[0,0,1]](−1)=[[1,−3,−11],[0,1,4],[0,0,1]]

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