Find the Inverse [[1,2,3],[2,5,7],[3,7,9]]

Problem

[[1,2,3],[2,5,7],[3,7,9]]

Solution

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

[[1,2,3,|,1,0,0],[2,5,7,|,0,1,0],[3,7,9,|,0,0,1]]

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

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

3. Eliminate the entries below and above the second pivot in the second column using (R_3)→(R_3)−(R_2) and (R_1)→(R_1)−2*(R_2)

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

4. Normalize the third row by multiplying by −1 ((R_3)→−(R_3).

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

5. Eliminate the entries above the third pivot in the third column using (R_1)→(R_1)−(R_3) and (R_2)→(R_2)−(R_3)

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

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

Final Answer

[[1,2,3],[2,5,7],[3,7,9]](−1)=[[4,−3,1],[−3,0,1],[1,1,−1]]

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