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

Problem

[[1,1,1],[0,2,3],[5,5,1]]

Solution

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

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

2. Eliminate the entry in the first column of the third row by performing the row operation (R_3)←(R_3)−5*(R_1)

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

3. Normalize the second row by dividing by 2 ((R_2)←1/2*(R_2) and the third row by dividing by -4 ((R_3)←−1/4*(R_3).

[[1,1,1,|,1,0,0],[0,1,1.5,|,0,0.5,0],[0,0,1,|,1.25,0,−0.25]]

4. Eliminate the entries above the leading 1 in the third column using (R_2)←(R_2)−1.5*(R_3) and (R_1)←(R_1)−(R_3)

[[1,1,0,|,−0.25,0,0.25],[0,1,0,|,−1.875,0.5,0.375],[0,0,1,|,1.25,0,−0.25]]

5. Eliminate the entry above the leading 1 in the second column using (R_1)←(R_1)−(R_2)

[[1,0,0,|,1.625,−0.5,−0.125],[0,1,0,|,−1.875,0.5,0.375],[0,0,1,|,1.25,0,−0.25]]

6. Convert the decimal values back into fractions to express the final inverse matrix.

1.625=13/8

−1.875=−15/8

0.375=3/8

Final Answer

[[1,1,1],[0,2,3],[5,5,1]](−1)=[[13/8,−1/2,−1/8],[−15/8,1/2,3/8],[5/4,0,−1/4]]

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