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

Problem

[[1,3,3],[1,4,3],[1,3,4]](−1)

Solution

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

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

2. Perform row operations to create zeros in the first column below the first pivot. Subtract row 1 from row 2 ((R_2)−(R_1)→(R_2) and subtract row 1 from row 3 ((R_3)−(R_1)→(R_3).

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

3. Eliminate the entries above the pivots in the second and third columns to reach reduced row echelon form. Subtract 3 times row 2 from row 1 ((R_1)−3*(R_2)→(R_1).

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

4. Subtract 3 times row 3 from row 1 ((R_1)−3*(R_3)→(R_1) to finish isolating the identity matrix on the left.

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

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

Final Answer

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

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