Rank of Matrix

The Rank of a Matrix is a fundamental concept in linear algebra that tells you how much “information” or “dimensionality” a matrix preserves. At its core, the rank represents the number of linearly independent rows or columns in a matrix. It is the dimension of the vector space generated by its columns (or rows).

https://www.mathsisfun.com/algebra/matrix-rank.html

Formal Definition

For a matrix A of size m×n:

• Column Rank: The maximum number of linearly independent column vectors in A.
• Row Rank: The maximum number of linearly independent row vectors in A.

Key Theorem: The Column Rank and Row Rank are always equal. We simply call this number the Rank of A, denoted as rank(A) or ρ(A).
https://en.wikipedia.org/wiki/Rank_(linear_algebra)

How to Find the Rank

The most reliable method to find the rank is to reduce the matrix to Row Echelon Form (REF) using Gaussian elimination.

Step-by-Step:

1. Perform row operations (swap rows, multiply by non-zero scalars, add multiples of rows) to get zeros below the main diagonal pivots.
2. Count the number of non-zero rows remaining.
3. This count is the rank.

Example: Find the rank of matrix A:

1. Observe that Row 2 (R2​) is exactly 2×R1​.
2. Subtract 2×R1​ from R2​.
3. Swap R2​ and R3​ to put the zero row at the bottom (Echelon Form).​​
4. There are 2 non-zero rows. Therefore, rank(A)=2.

Geometrical Meaning of Rank of 3×3 Matrix | What is Rank? (Linear Algebra) (Part 2)