Find the Nullity [[1,-1,3],[6,7,-3],[9,4,6]]

Problem

Nullity*[[1,−1,3],[6,7,−3],[9,4,6]]

Solution

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

A=[[1,−1,3],[6,7,−3],[9,4,6]]

2. Perform row operations to transform the matrix into row-echelon form. Subtract 6 times the first row from the second row ((R_2)−6*(R_1)⇒(R_2).

[[1,−1,3],[0,13,−21],[9,4,6]]

3. Continue row reduction by subtracting 9 times the first row from the third row ((R_3)−9*(R_1)⇒(R_3).

[[1,−1,3],[0,13,−21],[0,13,−21]]

4. Simplify the third row by subtracting the second row from the third row ((R_3)−(R_2)⇒(R_3).

[[1,−1,3],[0,13,−21],[0,0,0]]

5. Identify the rank of the matrix by counting the number of non-zero rows in the row-echelon form.

Rank(A)=2

6. Apply the Rank-Nullity Theorem, which states that for an n×n matrix, Rank(A)+Nullity(A)=n Here n=3

Nullity(A)=3−2

7. Calculate the final value.

Nullity(A)=1

Final Answer

Nullity*[[1,−1,3],[6,7,−3],[9,4,6]]=1

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