Find the Inverse [[4,-2,6],[0,1,-3],[0,0,6]]

Problem

[[4,−2,6],[0,1,−3],[0,0,6]](−1)

Solution

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

[[4,−2,6,|,1,0,0],[0,1,−3,|,0,1,0],[0,0,6,|,0,0,1]]

2. Normalize the third row by multiplying (R_3) by 1/6 to create a leading 1

[[4,−2,6,|,1,0,0],[0,1,−3,|,0,1,0],[0,0,1,|,0,0,1/6]]

3. Eliminate the entries above the leading 1 in the third column using the operations (R_2)←(R_2)+3*(R_3) and (R_1)←(R_1)−6*(R_3)

[[4,−2,0,|,1,0,−1],[0,1,0,|,0,1,1/2],[0,0,1,|,0,0,1/6]]

4. Eliminate the entry above the leading 1 in the second column using the operation (R_1)←(R_1)+2*(R_2)

[[4,0,0,|,1,2,0],[0,1,0,|,0,1,1/2],[0,0,1,|,0,0,1/6]]

5. Normalize the first row by multiplying (R_1) by 1/4 to obtain the identity matrix on the left.

[[1,0,0,|,1/4,1/2,0],[0,1,0,|,0,1,1/2],[0,0,1,|,0,0,1/6]]

Final Answer

[[4,−2,6],[0,1,−3],[0,0,6]](−1)=[[1/4,1/2,0],[0,1,1/2],[0,0,1/6]]

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