Definition:Rank/Matrix

Definition

Let $K$ be a field.

Let $\mathbf A$ be an $m \times n$ matrix over $K$.

Definition 1

The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the dimension of the subspace of $K^m$ generated by the columns of $\mathbf A$.

That is, it is the dimension of the column space of $\mathbf A$.

Definition 2

Let $\mathbf A$ be converted to echelon form $\mathbf B$.

Let $\mathbf B$ have exactly $k$ non-zero rows.

Then the rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is $k$.

Definition 3

The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the largest number of elements in a linearly independent set of rows of $\mathbf A$.

Definition 4

The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the largest number of elements in a linearly independent set of columns of $\mathbf A$.

Definition 5

The rank of $\mathbf A$, denoted $\map \rho {\mathbf A}$, is the order of the largest non-zero determinant that can be extracted from $\mathbf A$ by selecting rows and columns.

Also known as

The rank of a matrix can also be referred to as:

its row rank

its column rank.

Some sources denote the rank of a matrix $\mathbf A$ as:

$\map {\mathrm {rk} } {\mathbf A}$

Examples

Arbitrary Matrix $1$

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

The rank of $\mathbf A$ is $3$.

Arbitrary Matrix $2$

Let $\mathbf A = \begin {bmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \end {bmatrix}$

The rank of $\mathbf A$ is $2$.

Arbitrary Matrix $3$

Let $\mathbf A = \begin {bmatrix} 1 & 2 & 3 & 5 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 1 \end {bmatrix}$

The rank of $\mathbf A$ is $3$.

Arbitrary Matrix $4$

Let $\mathbf A = \begin {bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 3 & 3 \end {bmatrix}$

The rank of $\mathbf A$ is $2$.

Arbitrary Matrix $5$

Let $\mathbf A = \begin {bmatrix} -1 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ -1 & -2 & -3 & -4 & -5 \end {bmatrix}$

The rank of $\mathbf A$ is $2$.

Also see

• Equivalence of Definitions of Rank of Matrix

• Results about the rank of a matrix can be found here.