Find the Nullity [[1,0,0,-3],[0,1,0,-1],[0,0,1,2]]

Problem

nullity*[[1,0,0,−3],[0,1,0,−1],[0,0,1,2]]

Solution

1. Identify the dimensions of the matrix. The given matrix A has m=3 rows and n=4 columns.

2. Determine the rank of the matrix. The matrix is already in reduced row echelon form (RREF). Since there are 3 leading ones (pivots), the rank of the matrix is 3

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

rank(A)+nullity(A)=n

4. Substitute the known values into the theorem equation to solve for the nullity.

3+nullity(A)=4

5. Calculate the final value by subtracting the rank from the number of columns.

nullity(A)=4−3

nullity(A)=1

Final Answer

nullity*[[1,0,0,−3],[0,1,0,−1],[0,0,1,2]]=1

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