Find the Nullity [[1,0,2/9],[0,1,-4/9],[0,0,0]]

Problem

Nullity*[[1,0,2/9],[0,1,−4/9],[0,0,0]]

Solution

1. Identify the dimensions of the matrix. The given matrix A is a 3×3 matrix, so the number of columns n is 3

2. Determine the rank of the matrix. The matrix is already in reduced row echelon form (RREF). The number of non-zero rows (or pivot columns) is 2

3. Apply the Rank-Nullity Theorem. The theorem states that for an m×n matrix, the rank plus the nullity equals the number of columns n

rank(A)+nullity(A)=n

4. Substitute the known values into the formula.

2+nullity(A)=3

5. Solve for the nullity.

nullity(A)=3−2

nullity(A)=1

Final Answer

Nullity*[[1,0,2/9],[0,1,−4/9],[0,0,0]]=1

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