Matrix

Definition

Let $R$ be a ring, for example a field, and let $m, n \ge 0$ be integers. An $m \times n$ matrix over $R$ is a family $A = (a_{ij})$ of elements $a_{ij} \in R$, indexed by the pairs $(i, j)$ with $1 \le i \le m$ and $1 \le j \le n$. The element $a_{ij}$ is the entry of $A$ in row $i$ and column $j$. The notation for $A$ is a rectangular array with $m$ rows and $n$ columns: $$A = \begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}.$$ The set of all $m \times n$ matrices over $R$ is written $M_{m \times n}(R)$ or $R^{m \times n}$.