Find the Inverse [[1000,100,10],[300,20,1],[60,2,0]]

Problem

[[1000,100,10],[300,20,1],[60,2,0]]

Solution

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

[[1000,100,10,|,1,0,0],[300,20,1,|,0,1,0],[60,2,0,|,0,0,1]]

2. Divide the first row by 10 to simplify the numbers.

[[100,10,1,|,0.1,0,0],[300,20,1,|,0,1,0],[60,2,0,|,0,0,1]]

3. Eliminate the first column entries below the pivot by performing (R_2)−3*(R_1)⇒(R_2) and (R_3)−0.6*(R_1)⇒(R_3)

[[100,10,1,|,0.1,0,0],[0,−10,−2,|,−0.3,1,0],[0,−4,−0.6,|,−0.06,0,1]]

4. Eliminate the second column entry below the pivot by performing (R_3)−0.4*(R_2)⇒(R_3)

[[100,10,1,|,0.1,0,0],[0,−10,−2,|,−0.3,1,0],[0,0,0.2,|,0.06,−0.4,1]]

5. Normalize the third row by multiplying by 5 to make the pivot 1

[[100,10,1,|,0.1,0,0],[0,−10,−2,|,−0.3,1,0],[0,0,1,|,0.3,−2,5]]

6. Eliminate the third column entries above the pivot by performing (R_2)+2*(R_3)⇒(R_2) and (R_1)−(R_3)⇒(R_1)

[[100,10,0,|,−0.2,2,−5],[0,−10,0,|,0.3,−3,10],[0,0,1,|,0.3,−2,5]]

7. Eliminate the second column entry above the pivot by performing (R_1)+(R_2)⇒(R_1)

[[100,0,0,|,0.1,−1,5],[0,−10,0,|,0.3,−3,10],[0,0,1,|,0.3,−2,5]]

8. Divide the first row by 100 and the second row by −10 to obtain the identity matrix on the left.

[[1,0,0,|,0.001,−0.01,0.05],[0,1,0,|,−0.03,0.3,−1],[0,0,1,|,0.3,−2,5]]

Final Answer

[[1000,100,10],[300,20,1],[60,2,0]](−1)=[[0.001,−0.01,0.05],[−0.03,0.3,−1],[0.3,−2,5]]

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