Find Rank and Nullity of a Matrix Calculator With Variables

Matrix Rank and Nullity:

\[ \text{Rank}(A) = \text{Dim}(\text{Row Space of } A) \] \[ \text{Nullity}(A) = \text{Dim}(\text{Kernel of } A) \] \[ \text{Rank}(A) + \text{Nullity}(A) = n \text{ (for an } m \times n \text{ matrix)} \]

Rank:

Nullity:

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1. What Are Rank and Nullity?

The rank of a matrix is the dimension of its row space (or column space), while the nullity is the dimension of its kernel (null space). Together, they satisfy the Rank-Nullity Theorem: Rank(A) + Nullity(A) = n, where n is the number of columns in matrix A.

2. How the Calculator Works

The calculator analyzes the matrix structure and identifies variables that affect the rank and nullity:

\[ \text{Rank}(A) = \text{Number of linearly independent rows} \] \[ \text{Nullity}(A) = \text{Dimension of solution space of } A\mathbf{x} = \mathbf{0} \]

For matrices with variables, the calculator shows how the rank and nullity depend on these variables.

3. Importance of Rank and Nullity

Details: Rank determines the number of linearly independent equations in a system, while nullity indicates the number of free variables in solutions. They are fundamental in solving linear systems and understanding linear transformations.

4. Using the Calculator

Tips: Enter the matrix with one row per line, elements separated by spaces. Use letters for variables. The calculator will show how rank and nullity depend on these variables.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between row rank and column rank?
A: For any matrix, row rank equals column rank, so we just refer to "rank".

Q2: How do variables affect rank and nullity?
A: Depending on variable values, rows may become linearly dependent, reducing rank and increasing nullity.

Q3: What's the maximum possible rank for an m×n matrix?
A: The maximum rank is min(m, n) - the smaller of row or column count.

Q4: When does a matrix have nullity zero?
A: When the matrix has full column rank (all columns are linearly independent).

Q5: How is this related to solving Ax=b?
A: The nullity gives the dimension of the homogeneous solution space, while rank determines if solutions exist.