Matrix¶

Definition¶

A matrix is a rectangular array of elements of a field, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has size $m\times n$$n$.

Two matrices are equal if and only if they have the same size and all corresponding entries are equal. A square matrix is a matrix with the same number of rows and columns.

Interpretation¶

Matrices store coefficients, describe linear maps in coordinates, and organize data in a way that supports algebraic operations. A matrix is a table of numbers in which the position of each entry carries meaning: its row and its column.

The figure shows a $3\times4$ matrix: $m=3$ rows and $n=4$ columns, always counted in that order. The green entry $a_{23}$ lies where row $2$ crosses column $3$; the first index gives the row and the second the column, so $a_{32}$ would be a different position. The size is read "$3$ by $4$".

What rows and columns mean depends on the use. In a linear system in standard form, each row holds the coefficients of one equation and each column belongs to one unknown. For the map $\boldsymbol{x}\mapsto\boldsymbol{A}\boldsymbol{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$, column $j$ is the image of the $j$th standard basis vector, so the number of columns is the number of input coordinates. In a data table, rows often correspond to observations and columns to variables.

A common slip is to treat a matrix as just a collection of numbers. The arrangement matters: $\begin{pmatrix}1&2\end{pmatrix}$ and $\begin{pmatrix}1\\2\end{pmatrix}$ contain the same numbers but are different matrices, of sizes $1\times2$ and $2\times1$$1$. Special shapes are studied on their own pages, such as the identity matrix and symmetric matrices, which equal their transpose.

Key Results¶

• Rows are listed first and columns second: an $m\times n$ matrix has $m$ rows and $n$ columns.
• A matrix can be viewed as a list of vertical or horizontal vectors; an $n\times1$ matrix is a column vector and a $1\times n$ matrix is a row vector.
• Matrices of different sizes cannot be equal; matrices of the same size are equal exactly when their corresponding entries match.

Examples¶

• $\begin{pmatrix}1&2\\3&4\end{pmatrix}$ is a $2\times2$ square matrix.
• $\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$ is a $2\times3$ matrix and is not square.

Prerequisites¶

• field

Related Concepts¶

• Vector
• Linear Map

Sources¶

• Math II: Linear Algebra · Week 3

Keywords: matrices, square matrix, rectangular matrix, order of a matrix, matrix equality, equal matrices Symbols: real numbers

Last updated: 20 October 2026 03:02