1. What is Rank and Nullity?

The rank of a matrix is the dimension of its column space (the number of linearly independent columns), while the nullity is the dimension of its null space (the number of free variables in the solution to Ax=0). According to the Rank-Nullity Theorem: Rank(A) + Nullity(A) = n, where n is the number of columns.

2. How Does the Calculator Work?

The calculator uses Gaussian elimination to find the rank of the matrix and then applies the Rank-Nullity Theorem:

Where:

• \( \text{Rank}(A) \) — Number of linearly independent columns
• \( \text{Nullity}(A) \) — Dimension of null space
• \( n \) — Number of columns in matrix A

3. Importance of Rank and Nullity

Details: Rank and nullity are fundamental concepts in linear algebra that help determine the solvability of linear systems, understand the structure of linear transformations, and analyze the properties of matrices.

4. Using the Calculator

Tips: Enter your matrix using comma separation for row elements and semicolon separation for rows. For example: "1,2,3;4,5,6;7,8,9" for a 3x3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the Rank-Nullity Theorem?
A: The theorem states that for any matrix A, Rank(A) + Nullity(A) = n, where n is the number of columns.

Q2: What does rank tell us about a matrix?
A: Rank indicates the number of linearly independent columns/rows and determines the dimension of the column/row space.

Q3: When is nullity zero?
A: Nullity is zero when the matrix has full column rank, meaning its columns are linearly independent.

Q4: Can rank exceed the number of columns?
A: No, rank cannot exceed the number of columns or the number of rows.

Q5: How are rank and invertibility related?
A: A square matrix is invertible if and only if it has full rank (rank = number of columns = number of rows).