Rank¶

Definition¶

The rank of a matrix is the number of nonzero rows in its reduced row echelon form.

Full row, full column, and full rank¶

For an $m\times n$ matrix $\boldsymbol{A}$ :

$\boldsymbol{A}$ has full row rank if $\operatorname{rank}(\boldsymbol{A})=m$ $m$ . This corresponds to the rows being linearly independent, so that $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ has at least one solution for every right-hand side $\boldsymbol{b}\in\mathbb{R}^m$ $R^{m}$ .
$\boldsymbol{A}$ has full column rank if $\operatorname{rank}(\boldsymbol{A})=n$ $n$ . This corresponds to columns being linearly independent, so $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ has at most one solution.
A square matrix has full rank if $\operatorname{rank}(\boldsymbol{A})=n$ $n$ ; this is equivalent to invertibility and to a nonzero determinant.

Interpretation¶

Rank measures how many independent pieces of information a matrix contains. It is the number of linearly independent columns, which equals the number of independent rows and the number of pivots left after row reduction.

Two panels: two column vectors pointing in different directions, which span the plane and give rank 2, and two parallel column vectors, which span only a line and give rank 1.

On the left, $\boldsymbol{A}=\begin{pmatrix}2.4&0.8\\0.6&2.0\end{pmatrix}$ has independent columns spanning $\mathbb{R}^2$ , so its rank is $2$ . On the right, $\boldsymbol{A}=\begin{pmatrix}2.4&1.2\\1.2&0.6\end{pmatrix}$ has proportional columns, $\boldsymbol{a}_2=\tfrac{1}{2}\boldsymbol{a}_1$ $\frac{1}{2} a_{1}$ , spanning only a line, so its rank is $1$ and its determinant is zero.

Rank counts the dimensions of the current image of $\boldsymbol{M}_t=(1-t)\boldsymbol{I}+t\boldsymbol{A}$ $t A$ . Rank two gives the whole plane and kernel $\{\boldsymbol{0}\}$ ; rank one gives a line image and line kernel; rank zero gives image $\{\boldsymbol{0}\}$ and the whole plane as kernel. At $t=1$ this is the target $\boldsymbol{A}$ .

Row operations do not change the solutions of $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ $0$ , so they keep every linear relation between the columns. In the echelon form the pivot columns are independent and every other column is a combination of them, and the same holds for $\boldsymbol{A}$ itself. So the number of pivots is the dimension of the range of $\boldsymbol{x}\mapsto\boldsymbol{A}\boldsymbol{x}$ $A x$ , the span of the columns.

For a system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ with $n$ unknowns, the system is solvable if and only if the augmented matrix $(\boldsymbol{A}\mid\boldsymbol{b})$ has the same rank as $\boldsymbol{A}$ , and then the solutions have $n-\operatorname{rank}(\boldsymbol{A})$ free variables. That number is the dimension of the kernel, by the fundamental theorem of linear maps.

A common slip is to count the nonzero rows of $\boldsymbol{A}$ itself rather than of its echelon form: $\begin{pmatrix}1&2\\2&4\end{pmatrix}$ has two nonzero rows but rank $1$ , because the second row is twice the first.

Key Results¶

The full row, full column, and full rank cases are special cases of rank.
The rank equals the dimension of the range of the corresponding linear map.
For an $m\times n$ matrix, $\operatorname{rank}(\boldsymbol{A})\leq m$ and $\operatorname{rank}(\boldsymbol{A})\leq n$ $n$ .

Examples¶

A zero matrix has rank $0$ .
$\boldsymbol{I}_n$ has rank $n$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 4
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: matrix rank Keywords: full rank, full row rank, full column rank Symbols: element of, real numbers

Last updated: 27 October 2026 11:52