1. What is Nullity of a Matrix?

The nullity of a matrix is the dimension of the null space (or kernel) of the matrix. It represents the number of linearly independent solutions to the homogeneous equation \( A\mathbf{x} = \mathbf{0} \).

2. How Does the Calculator Work?

The calculator uses the Rank-Nullity Theorem:

Where:

• \( n \) — Number of columns in the matrix
• \( \text{Rank}(A) \) — Rank of matrix A (number of linearly independent rows or columns)

Explanation: The rank-nullity theorem connects the dimensions of the column space and null space of a matrix.

3. Importance of Nullity Calculation

Details: Nullity helps determine the number of free variables in a system of linear equations and is fundamental in linear algebra for understanding matrix properties and solving systems of equations.

4. Using the Calculator

Tips: Enter the number of columns in the matrix and its rank. The rank must be less than or equal to both the number of rows and columns of the matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between nullity and rank?
A: According to the rank-nullity theorem, the sum of the rank and nullity of a matrix equals the number of columns.

Q2: Can nullity be zero?
A: Yes, when the matrix has full column rank (rank equals number of columns), the nullity is zero.

Q3: What does a high nullity indicate?
A: A high nullity indicates many linearly independent solutions to the homogeneous equation, meaning the matrix has many linear dependencies.

Q4: How is nullity related to invertibility?
A: A square matrix is invertible if and only if its nullity is zero (it has full rank).

Q5: Does nullity depend on the field?
A: Yes, the nullity may change if the matrix is considered over different fields (e.g., real vs complex numbers).