1. What Is the Rank of a Matrix?
The rank of a matrix is a number that tells us how much useful or non-redundant information the matrix contains. More precisely, it is the maximum number of linearly independent rows or columns in the matrix.
In simpler language: the rank counts how many rows (or columns) are truly different from each other and not just multiples or combinations of one another.
2. Definition of Rank
The rank of a matrix \( A \) is:
- The number of non-zero rows in its row-echelon form.
- The number of linearly independent rows or columns.
Both row rank and column rank are always equal, so we simply call it the rank.
3. How to Find the Rank of a Matrix
The usual method to find rank is to use elementary row transformations to convert the matrix into:
- Row Echelon Form (REF), or
- Reduced Row Echelon Form (RREF)
Then, count how many rows are non-zero. That count is the rank.
3.1. What Is a Non-Zero Row?
A row is non-zero if it contains at least one element that is not zero.
Example:
- \( [0\; 0\; 5] \) → non-zero row
- \( [0\; 0\; 0] \) → zero row
4. Example 1: Finding Rank Using Row Echelon Form
Let:
A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}
Apply row operations:
- \( R_2 \rightarrow R_2 - 2R_1 \)
- \( R_3 \rightarrow R_3 - 3R_1 \)
A \sim \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
4.1. Conclusion
Only one non-zero row remains.
So, the rank is:
\( \text{rank}(A) = 1 \)
5. Example 2: Another Rank Calculation
Consider:
B = \begin{bmatrix} 2 & 3 \\ 4 & 7 \end{bmatrix}
Apply:
R_2 \rightarrow R_2 - 2R_1
B \sim \begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix}
There are two non-zero rows.
\text{rank}(B) = 2
6. Maximum Possible Rank
The maximum rank a matrix can have equals the smaller of:
- number of rows
- number of columns
This is because you can't have more independent rows than total rows.
Example:
- A \( 3 \times 5 \) matrix → rank ≤ 3
- A \( 4 \times 2 \) matrix → rank ≤ 2
7. When Is a Matrix Full Rank?
A matrix is called full rank when its rank equals the smallest of its dimensions.
Examples:
- A \( 3 \times 3 \) matrix with rank 3 → full rank.
- A \( 2 \times 4 \) matrix with rank 2 → full row rank.
Full-rank matrices do not have redundant rows or columns.
8. Why Rank Matters
Rank is important because it tells us:
- whether a system of linear equations has unique, infinite, or no solutions
- whether a matrix is invertible (rank must be full)
- how many independent directions a matrix captures
- the dimension of row space and column space
Rank is a fundamental idea that appears throughout linear algebra and applications.