1. What is Rank and Nullity?

The rank of a matrix is the dimension of its column space (or row space), representing the maximum number of linearly independent column vectors. The nullity is the dimension of the null space, representing the number of free variables in the solution to Ax=0.

2. How Does the Calculator Work?

The calculator uses the fundamental theorem of linear algebra:

Where \( n \) is the number of columns in the matrix. The rank is calculated by performing Gaussian elimination to find the number of pivot columns in the row echelon form.

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 the matrix with rows separated by semicolons and elements within each row separated by commas. For example: "1,2,3;4,5,6;7,8,9" for a 3x3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between rank and nullity?
A: The rank-nullity theorem states that for any matrix, rank + nullity = number of columns.

Q2: What does a nullity of zero mean?
A: A nullity of zero means the matrix has a trivial null space (only the zero vector), indicating the columns are linearly independent.

Q3: Can rank exceed the number of rows?
A: No, the rank cannot exceed the minimum of the number of rows and columns.

Q4: What is the significance of full rank?
A: A matrix has full rank if its rank equals the minimum of its number of rows and columns, indicating maximum possible linear independence.

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