Find the Nullity [[2,2,6],[-4,5,-1],[3,7,1]]

Problem

Nullity*[[2,2,6],[−4,5,−1],[3,7,1]]

Solution

1. Set up the matrix A and the equation A*x=0 to find the null space.

A=[[2,2,6],[−4,5,−1],[3,7,1]]

2. Perform row operations to transform the matrix into row-echelon form. Divide the first row by 2

[[1,1,3],[−4,5,−1],[3,7,1]]

3. Eliminate the entries below the first pivot by adding 4 times the first row to the second row, and subtracting 3 times the first row from the third row.

[[1,1,3],[0,9,11],[0,4,−8]]

4. Simplify the third row by dividing by 4

[[1,1,3],[0,9,11],[0,1,−2]]

5. Swap the second and third rows to make calculations easier.

[[1,1,3],[0,1,−2],[0,9,11]]

6. Eliminate the entry below the second pivot by subtracting 9 times the second row from the third row.

[[1,1,3],[0,1,−2],[0,0,29]]

7. Determine the rank by counting the number of non-zero rows (pivots) in the row-echelon form.

Rank(A)=3

8. Apply the Rank-Nullity Theorem, which states that Rank(A)+Nullity(A)=n where n is the number of columns.

3+Nullity(A)=3

9. Solve for the nullity.

Nullity(A)=0

Final Answer

Nullity*[[2,2,6],[−4,5,−1],[3,7,1]]=0

Want more problems? Check here!