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

Problem

[[1,−2,3],[0,1,2],[0,0,2]]

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],[0,1,2,|,0,1,0],[0,0,2,|,0,0,1]]

2. Normalize the third row by dividing (R_3) by 2 to create a leading 1

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

3. Eliminate the entries above the leading 1 in the third column by performing (R_2)⇒(R_2)−2*(R_3) and (R_1)⇒(R_1)−3*(R_3)

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

4. Eliminate the entry above the leading 1 in the second column by performing (R_1)⇒(R_1)+2*(R_2)

[[1,0,0,|,1,2,−3.5],[0,1,0,|,0,1,−1],[0,0,1,|,0,0,0.5]]

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

Final Answer

[[1,−2,3],[0,1,2],[0,0,2]](−1)=[[1,2,−3.5],[0,1,−1],[0,0,0.5]]

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