Find the Nullity [[1,4,0,3,0],[0,0,1,-5,0],[0,0,0,0,4]]

Problem

nullity*[[1,4,0,3,0],[0,0,1,−5,0],[0,0,0,0,4]]

Solution

1. Identify the dimensions of the matrix. The given matrix A has m=3 rows and n=5 columns.

2. Determine the rank of the matrix. The matrix is already in row echelon form. The pivot elements are the first non-zero entries in each row, located at (1,1) (2,3) and (3,5)

3. Count the number of pivots. Since there are 3 pivots, the rank of the matrix is 3.

rank(A)=3

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

rank(A)+nullity(A)=n

5. Substitute the known values into the formula.

3+nullity(A)=5

6. Solve for the nullity.

nullity(A)=5−3

nullity(A)=2

Final Answer

nullity*[[1,4,0,3,0],[0,0,1,−5,0],[0,0,0,0,4]]=2

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