Find the Inverse [[1,1,1],[3,5,4],[3,6,5]]

Problem

[[1,1,1],[3,5,4],[3,6,5]]

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],[3,5,4,|,0,1,0],[3,6,5,|,0,0,1]]

2. Eliminate the entries in the first column below the pivot by performing row operations (R_2)−3*(R_1)⇒(R_2) and (R_3)−3*(R_1)⇒(R_3)

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

3. Create a pivot of 1 in the second row by performing 1/2*(R_2)⇒(R_2)

[[1,1,1,|,1,0,0],[0,1,0.5,|,−1.5,0.5,0],[0,3,2,|,−3,0,1]]

4. Eliminate the entries in the second column around the pivot by performing (R_1)−(R_2)⇒(R_1) and (R_3)−3*(R_2)⇒(R_3)

[[1,0,0.5,|,2.5,−0.5,0],[0,1,0.5,|,−1.5,0.5,0],[0,0,0.5,|,1.5,−1.5,1]]

5. Create a pivot of 1 in the third row by performing 2*(R_3)⇒(R_3)

[[1,0,0.5,|,2.5,−0.5,0],[0,1,0.5,|,−1.5,0.5,0],[0,0,1,|,3,−3,2]]

6. Eliminate the entries in the third column above the pivot by performing (R_1)−0.5*(R_3)⇒(R_1) and (R_2)−0.5*(R_3)⇒(R_2)

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

Final Answer

[[1,1,1],[3,5,4],[3,6,5]](−1)=[[1,1,−1],[−3,2,−1],[3,−3,2]]

Want more problems? Check here!