1. What is Matrix Nullity?

The nullity of a matrix is the dimension of its null space (kernel). 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 fundamental theorem:

Where:

• \( A \) — Input matrix
• \( n \) — Number of columns in matrix
• \( \text{rank}(A) \) — Rank of matrix (number of linearly independent rows/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 is crucial in linear algebra for understanding solution spaces of linear systems, determining matrix invertibility, and analyzing linear transformations.

4. Using the Calculator

Tips: Enter matrix values separated by commas for columns and semicolons for rows. For example: "1,2,3;4,5,6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What does nullity = 0 mean?
A: A nullity of 0 means the matrix has full column rank and the only solution to \( A\mathbf{x} = \mathbf{0} \) is the trivial solution.

Q2: How is nullity related to linear independence?
A: The nullity gives the number of linearly dependent columns in the matrix (when nullity > 0).

Q3: Can nullity exceed the number of columns?
A: No, nullity is always ≤ n (number of columns) since rank ≥ 0.

Q4: What's the nullity of an invertible matrix?
A: For an n×n invertible matrix, rank = n, so nullity = 0.

Q5: How does nullity relate to eigenvalues?
A: The nullity of \( A - \lambda I \) gives the geometric multiplicity of eigenvalue λ.