Find the Inverse [[1,0,9],[0,1,6],[0,0,3]]

Problem

[[1,0,9],[0,1,6],[0,0,3]](−1)

Solution

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

[[1,0,9,|,1,0,0],[0,1,6,|,0,1,0],[0,0,3,|,0,0,1]]

2. Divide the third row by 3 to create a leading 1 in the third column.

[[1,0,9,|,1,0,0],[0,1,6,|,0,1,0],[0,0,1,|,0,0,1/3]]

3. Eliminate the entry in the second row, third column by performing the row operation (R_2)−6*(R_3)⇒(R_2)

[[1,0,9,|,1,0,0],[0,1,0,|,0,1,−2],[0,0,1,|,0,0,1/3]]

4. Eliminate the entry in the first row, third column by performing the row operation (R_1)−9*(R_3)⇒(R_1)

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

5. Identify the resulting matrix on the right side as the inverse matrix A(−1)

Final Answer

[[1,0,9],[0,1,6],[0,0,3]](−1)=[[1,0,−3],[0,1,−2],[0,0,1/3]]

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