The formula to calculate the rank and nullity of a matrix is:
\[ \text{Rank}(A) + \text{Nullity}(A) = n \]
Where:
Rank and nullity are concepts in linear algebra that describe certain properties of a linear transformation or a matrix. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the image or output of the transformation. The nullity of a matrix, on the other hand, is the dimension of the kernel or null space of the transformation, which is the set of all vectors that get mapped to the zero vector. The rank-nullity theorem states that the sum of the rank and the nullity of a matrix equals the number of its columns.
Let's assume the following values:
Step 1: Apply the rank-nullity theorem:
\[ \text{Nullity}(A) = n - \text{Rank}(A) \]
Step 2: Substitute the values:
\[ \text{Nullity}(A) = 5 - 3 = 2 \]
The Nullity of the Matrix is 2.