Rank of a Matrix

What is rank?

The rank of an $m \times n$ matrix $A$, written $\operatorname{rank}(A)$, is the dimension of its column space — the number of linearly independent columns. Equivalently, it's:

• The number of pivots in any row-echelon form of $A$.
• The dimension of the row space (row rank $=$ column rank — always).
• The number of non-zero rows in the RREF.

Rank satisfies $0 \le \operatorname{rank}(A) \le \min(m, n)$.

Step-by-step

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$

Step 1 — Row reduce.

$R_2 \to R_2 - 4 R_1$: $(4, 5, 6) - 4(1, 2, 3) = (0, -3, -6)$. $R_3 \to R_3 - 7 R_1$: $(7, 8, 9) - 7(1, 2, 3) = (0, -6, -12)$.

$\begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \end{pmatrix}$

$R_3 \to R_3 - 2 R_2$: $(0, -6, -12) - 2(0, -3, -6) = (0, 0, 0)$.

$\begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 0 \end{pmatrix}$

Step 2 — Count pivots. Row 1 has a pivot in column 1; row 2 has a pivot in column 2; row 3 is zero (no pivot).

$\operatorname{rank}(A) = 2$

Since $A$ is $3 \times 3$ but rank 2, the matrix is rank-deficient: it has fewer pivots than its size. It is singular ($\det A = 0$) and not invertible.

A rank-2 story: what went wrong?

The original rows are arithmetic progressions: $(1,2,3), (4,5,6), (7,8,9)$. Each row is the previous one shifted by 3. In particular, $\mathbf{r}_3 = 2 \mathbf{r}_2 - \mathbf{r}_1$

One row is a linear combination of the others, so they span only a 2D subspace of $\mathbb{R}^3$.

Rank and properties

• $\operatorname{rank}(A) = n$ (full column rank) ⟺ columns are linearly independent ⟺ $A \mathbf{x} = \mathbf{0}$ has only the trivial solution.
• $\operatorname{rank}(A) = m$ (full row rank) ⟺ $A \mathbf{x} = \mathbf{b}$ is consistent for every $\mathbf{b}$.
• For a square $n \times n$ matrix: $\operatorname{rank}(A) = n$ ⟺ $A$ is invertible ⟺ $\det A \ne 0$.

How rank changes under operations

• Elementary row operations preserve rank.
• Transpose: $\operatorname{rank}(A^T) = \operatorname{rank}(A)$ — the row rank equals the column rank.
• Multiplication by invertible: $\operatorname{rank}(PA) = \operatorname{rank}(AP) = \operatorname{rank}(A)$ if $P$ is invertible.
• Sub-additivity: $\operatorname{rank}(A + B) \le \operatorname{rank}(A) + \operatorname{rank}(B)$.
• Multiplicativity: $\operatorname{rank}(AB) \le \min(\operatorname{rank}(A), \operatorname{rank}(B))$.

Common mistakes

• Counting rows instead of pivots. Zero rows in the reduced form contribute no pivot and no rank.
• Confusing rank with the number of non-zero entries. A matrix can be dense but have low rank (like $A$ above).
• Not being careful with identifying pivot columns vs. pivot rows. Pivots are at specific (row, column) positions, and the column index determines which column is a pivot column.

Try it in the visualization

Type entries into a $3 \times 3$ matrix. As you row-reduce, pivots light up; rank updates live; a small 3D plot shows the column span — flat plane if rank 2, full space if rank 3, line if rank 1.