Matrix Rank

What is Matrix Rank?

The rank of a matrix is a fundamental concept in linear algebra that measures the "nondegenerateness" of the system of linear equations represented by the matrix. It is defined as the maximum number of linearly independent row vectors in the matrix, or equivalently, the maximum number of linearly independent column vectors.

Formula and Meaning

There is no single formula for calculating matrix rank, but it can be determined through various methods:

• Gaussian elimination to row echelon form
• Counting the number of non-zero rows in row echelon form
• Calculating the dimension of the column space or row space

The rank of a matrix A is often denoted as rank(A) or r(A).

Calculation Steps

To calculate the rank of a matrix using Gaussian elimination:

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

Example

Let's calculate the rank of the following 3x3 matrix:

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}\]

Step 1: Convert to row echelon form

\[\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\]

Step 2: Count non-zero rows

There is only one non-zero row, so rank(A) = 1

Visual Representation

This diagram shows the row echelon form of the example matrix. The green cells represent non-zero entries, while red cells represent zero entries.