1. What is Matrix Rank?

The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors) in the matrix. It represents the dimension of the vector space spanned by its rows or columns.

2. How Rank is Calculated

The calculator uses Gaussian elimination to transform the matrix to row echelon form:

Steps:

1. Convert the matrix to row echelon form using elementary row operations
2. Count the number of non-zero rows in the resulting matrix

3. Importance of Matrix Rank

Applications: Matrix rank is fundamental in linear algebra for determining:

• The number of solutions to a system of linear equations
• The dimension of the column space or row space of a matrix
• Whether a matrix is invertible (full rank matrices are invertible)
• The number of linearly independent vectors in a set

4. Using the Calculator

Instructions: Enter your matrix using commas to separate elements within a row and semicolons to separate rows. Example: "1,2,3;4,5,6;7,8,9" represents a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the maximum possible rank of an m×n matrix?
A: The rank cannot exceed min(m,n). A matrix with rank equal to min(m,n) is said to have full rank.

Q2: What does rank zero mean?
A: A rank of zero means all elements of the matrix are zero (the zero matrix).

Q3: How does rank relate to determinants?
A: For square matrices, a matrix has full rank if and only if its determinant is non-zero.

Q4: Can rank be different for rows vs columns?
A: No, the row rank always equals the column rank for any matrix.

Q5: What's the rank of a non-square matrix?
A: The rank is the number of linearly independent rows or columns (whichever is smaller).