1. What Is Matrix Rank?

The rank of a matrix is defined as the maximum number of linearly independent rows or columns. It represents the dimension of the vector space generated by its rows or columns.

2. How Does The Calculator Work?

The calculator computes the rank by transforming the matrix to row echelon form using Gaussian elimination and counting the number of non-zero rows:

Explanation: The algorithm performs elementary row operations to reduce the matrix to row echelon form, where all non-zero rows are linearly independent.

3. Importance Of Matrix Rank

Details: Matrix rank is fundamental in linear algebra with applications in solving systems of linear equations, determining invertibility of matrices, and analyzing linear transformations. It's crucial in fields like engineering, physics, and data science.

4. Using The Calculator

Tips: Enter your matrix using comma-separated values for columns and semicolon-separated values for rows. For example: "1,2,3;4,5,6;7,8,9" represents a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What does rank tell us about a matrix?
A: The rank indicates the maximum number of linearly independent rows or columns, which determines the dimension of the column space and row space of the matrix.

Q2: What is the relationship between rank and invertibility?
A: A square matrix is invertible if and only if its rank equals its dimension (full rank).

Q3: Can a matrix have different row and column ranks?
A: No, the row rank and column rank of any matrix are always equal.

Q4: What is the maximum possible rank of an m×n matrix?
A: The maximum rank is min(m, n), the smaller of the row and column dimensions.

Q5: How does rank relate to solutions of linear systems?
A: For a system Ax = b, if rank(A) = rank([A|b]) = n (number of variables), there's a unique solution. If rank(A) = rank([A|b]) < n, there are infinitely many solutions.