What Is the Notation for a Zero Matrix?

A zero matrix is a matrix whose entries are all zero. Common notations include $0$, $\boldsymbol{0}$, and $O$; none is universal. When the size is not obvious from context, include it as a subscript, as in $0_{2\times 3}$ or $O_{2\times 3}$.

The dimensions matter: a $2\times 3$ zero matrix and a $3\times 2$ zero matrix contain only zeros, but they are different matrices. The notation should make clear which one is being used whenever that distinction matters.

Common notations

Notation	Typical use
$0$	A convenient shorthand when the matrix dimensions are clear from context
$\boldsymbol{0}$ or $\mathbf{0}$	Emphasizes that the zero is a vector or matrix rather than a scalar
$O$	A capital letter sometimes used for a zero matrix, especially in older texts or particular conventions
$0_{m\times n}$ or $O_{m\times n}$	Explicitly states that the zero matrix has $m$ rows and $n$ columns

For example, the $2\times 3$ zero matrix can be written as

$$0_{2\times 3}=\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}.$$

Some authors write the same matrix as $\boldsymbol{0}_{2\times 3}$ or $O_{2\times 3}$. Follow the convention used in your course or text, and define a less familiar symbol if you introduce it yourself.

Why the dimensions matter

For fixed dimensions, matrices form a vector space under entrywise addition and scalar multiplication. Its zero element is the matrix with those same dimensions and all entries zero. Thus, in the space $M_{2\times 3}(\mathbb{R})$, the additive identity is $0_{2\times 3}$.

The size also determines which operations are defined. If $A$ is a $2\times 3$ matrix, then $A+0_{2\times 3}$ is defined and equals $A$. But adding a $3\times 2$ matrix to $A$ is not defined. Likewise, matrix multiplication requires compatible dimensions: a $2\times 3$ matrix can be multiplied on the right by a $3\times 4$ matrix, producing a $2\times 4$ result.

Worked example

Let

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

so $A$ has size $2\times 3$. The matrix that acts as its additive identity must also be $2\times 3$:

$$A+0_{2\times 3} =\begin{bmatrix}1&-2&4\\0&3&5\end{bmatrix} +\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix} =\begin{bmatrix}1&-2&4\\0&3&5\end{bmatrix}=A.$$

By contrast, $0_{3\times 2}$ cannot be added to $A$, because their dimensions differ. The entries being zero does not make the sizes interchangeable.

For multiplication, take a $3\times 2$ zero matrix. Then

$$A0_{3\times 2}=0_{2\times 2},$$

because the product has the number of rows of $A$ and the number of columns of the second factor. The result is a $2\times 2$ zero matrix, not a $2\times 3$ one.

Zero matrix versus scalar zero

The symbol $0$ can mean the scalar zero, a zero vector, or a zero matrix. Usually the context tells you which object is meant. For instance, in a matrix equation $A+0=A$, the zero must be a matrix with the same dimensions as $A$. In an equation between real numbers, $0$ is a scalar.

When ambiguity could affect the argument, write the dimensions or use boldface. In a vector space of matrices, it is also common to call its additive identity simply $0$, because the space has already specified what kind of object that zero is.

Common mistakes

• Assuming every zero matrix has the same size. A zero matrix is an element of a particular matrix space, such as $M_{2\times 3}(\mathbb{R})$.
• Treating $O$ as a universal convention. Some books use $O$ for a zero matrix; others use it differently. Check the author’s definitions.
• Confusing a zero matrix with the identity matrix. The identity matrix $I_n$ has ones on its main diagonal and zeros elsewhere. It is not a zero matrix.
• Ignoring dimensions in products. Multiplication by a zero matrix gives a zero result of a size determined by the product’s dimensions, if the product is defined.

If a problem gives no special notation, writing $0_{m\times n}$ is a clear choice whenever the size is relevant. Otherwise, plain $0$ is standard shorthand once the dimensions are understood.