# Definition:Matrix/Element

## Definition

Let $\mathbf A$ be an $m \times n$ matrix over a set $S$.

The individual $m \times n$ elements of $S$ that go to form $\mathbf A = \sqbrk a_{m n}$ are known as the **elements of the matrix**.

The **element** at row $i$ and column $j$ is called **element $\tuple {i, j}$ of $\mathbf A$**, and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character.

If the indices are still more complicated coefficients and further clarity is required, then the form $a \tuple {i, j}$ can be used.

Note that the first subscript determines the row, and the second the column, of the matrix where the **element** is positioned.

## Also denoted as

Some sources prefer to use the uppercase form of the letter for the **matrix element**:

- $A_{i j}$

## Also known as

An **element of a matrix** is sometimes seen as **entry of a matrix**, or just **(matrix) entry**.

Earlier sources may be seen to use the words **constituent** or even **coordinate**, but these names have now been superseded.

## Sources

- 1954: A.C. Aitken:
*Determinants and Matrices*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions and Fundamental Operations of Matrices: $3$. The Notation of Matrices - 1980: A.J.M. Spencer:
*Continuum Mechanics*... (previous) ... (next): $2.1$: Matrices - 1982: A.O. Morris:
*Linear Algebra: An Introduction*(2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Definition $1.1$ - 1998: Richard Kaye and Robert Wilson:
*Linear Algebra*... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.1$ Matrices - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.3$: $m \times n$ matrices - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**entry**