Find the Nullity [[1,x,3],[-2,10,-9]]

Problem

Nullity*[[1,x,3],[−2,10,−9]]

Solution

1. Identify the dimensions of the matrix. The matrix A has m=2 rows and n=3 columns.

2. Determine the rank of the matrix. Since the first column [[1],[−2]] and the third column [[3],[−9]] are not multiples of each other, they are linearly independent. Therefore, the rank of the matrix is 2

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

4. Substitute the known values into the formula rank(A)+nullity(A)=n

2+nullity(A)=3

5. Solve for the nullity by subtracting the rank from the number of columns.

nullity(A)=3−2

nullity(A)=1

Final Answer

Nullity*[[1,x,3],[−2,10,−9]]=1

Want more problems? Check here!