Introduction

The rank of a matrix is one of the most important ideas in linear algebra.
It tells us how much useful information a matrix contains.

You already know how to compute RREF, so this article will show you how RREF reveals the rank and what the rank tells us about vectors, systems of equations, and dimensions.

What Is Rank?

At a high level:

• The rank of a matrix is the number of pivots in its RREF.
• Equivalently, it is the number of linearly independent rows or linearly independent columns.
• Rank measures how many directions of information the matrix contains.

A matrix with:

• High rank → lots of independent information
• Low rank → much of the information is redundant or dependent
• Rank 0 → the zero matrix (no information at all)

Why Rank Matters

Rank tells us:

• Whether a system of linear equations has no, one, or infinitely many solutions.
• Whether a set of vectors spans a line, a plane, or something higher-dimensional.
• Whether columns of a matrix contain redundant information.
• Whether a transformation “squashes” space into a lower dimension.

Some useful interpretations:

• Rank = number of independent rows
• Rank = number of independent columns
• Rank = dimension of the row space
• Rank = dimension of the column space

All of these numbers are equal.

Computing Rank Using RREF

To compute the rank of a matrix:

1. Convert the matrix to RREF.
2. Count the number of pivot columns (columns with a leading 1).
3. That count is the rank.

Example: $$\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{bmatrix} \longrightarrow \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}$$

• There are 2 pivots, so the matrix has rank 2.
• This means only two rows (or columns) contain independent information.

Interpreting Rank

Some common interpretations:

1. Rank and Linear Independence

• If the rank equals the number of columns, the columns are linearly independent.
• If the rank is smaller, some columns are combinations of others.

2. Rank and Solutions to Linear Systems

For a system $A x = b$:

• If $\text{rank}(A) < \text{rank}([A|b])$ → no solution
• If $\text{rank}(A) = \text{rank}([A|b]) = n$ → unique solution
• If $\text{rank}(A) = \text{rank}([A|b]) < n$ → infinitely many solutions

3. Rank and Dimension

Rank tells you the dimension of:

• The column space (span of columns)
• The row space (span of rows)

Examples

Example 1: A full-rank matrix

$$A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

• RREF is the same.
• Two pivots → rank 2.
• The columns span the entire plane.

Example 2: A low-rank matrix

$$B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$$

• Second row is twice the first.
• RREF has only one pivot.
• Rank = 1 → the matrix contains only one direction of information.

Calculator

Exercises

1. Compute the rank of $$\begin{bmatrix}1 & 2 \\ 3 & 6\end{bmatrix}$$

   Solution

   Matrix: $$\begin{bmatrix}1 & 2 \\ 3 & 6\end{bmatrix}$$ Row 2 is a multiple of Row 1 → rank = 1.

2. Compute the rank of $$\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 1\end{bmatrix}$$

   Solution

   Matrix: $$\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 1\end{bmatrix}$$ Two pivots → rank = 2.

3. Determine whether the columns of $$\begin{bmatrix}2 \\ 4\end{bmatrix},\ \begin{bmatrix}1 \\ 2\end{bmatrix}$$ are linearly independent.

   Solution

   Vectors $(2,4)$ and $(1,2)$ are multiples of each other → not independent.

4. Find the rank of the matrix $$\begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & 4\end{bmatrix}$$

   Solution

   Matrix: $$\begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 3 & 4\end{bmatrix}$$ RREF has two pivots → rank = 2.

5. True or false: A $3\times 3$ matrix with rank 1 has columns that all lie on a single line.

   Solution

   True.
   Rank 1 means all columns lie on a single line through the origin.

6. Compute the rank of $$\begin{bmatrix}0 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7\end{bmatrix}$$

   Solution

   Matrix: $$\begin{bmatrix}0 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7\end{bmatrix}$$ Two nonzero rows → rank = 2.

7. Determine whether the system $A x = 0$ has non‑trivial solutions when $$A = \begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & 6\end{bmatrix}$$

   Solution

   Matrix: $$\begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & 6\end{bmatrix}$$ Rank = 1 < number of columns (3) → the system has non‑trivial solutions.