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

Problem

[[1,0,−3],[2,−2,1],[0,−1,3]]

Solution

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

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

2. Eliminate the first entry in the second row by performing the row operation (R_2)←(R_2)−2*(R_1)

[[1,0,−3,|,1,0,0],[0,−2,7,|,−2,1,0],[0,−1,3,|,0,0,1]]

3. Swap the second and third rows to make the calculations easier, then multiply the new second row by −1 to get a leading 1.

[[1,0,−3,|,1,0,0],[0,−1,3,|,0,0,1],[0,−2,7,|,−2,1,0]]

(R_2)←−1*(R_2)

[[1,0,−3,|,1,0,0],[0,1,−3,|,0,0,−1],[0,−2,7,|,−2,1,0]]

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

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

5. Eliminate the entries in the third column of the first and second rows by performing (R_1)←(R_1)+3*(R_3) and (R_2)←(R_2)+3*(R_3)

(R_1)←(R_1)+3*(R_3)

[[1,0,0,|,−5,3,−6],[0,1,−3,|,0,0,−1],[0,0,1,|,−2,1,−2]]

(R_2)←(R_2)+3*(R_3)

[[1,0,0,|,−5,3,−6],[0,1,0,|,−6,3,−7],[0,0,1,|,−2,1,−2]]

6. Identify the resulting matrix on the right side as the inverse.

Final Answer

[[1,0,−3],[2,−2,1],[0,−1,3]](−1)=[[−5,3,−6],[−6,3,−7],[−2,1,−2]]

Want more problems? Check here!