Matrix

Definition 1. Let $R$ be a ring. An $m \times n$-matrix in $R$ ($m$, $n \in \N^+$) is a function from $\llbra m \rrbra \times \llbra n \rrbra$ to $R$.

The set of all $m \times n$-matrices in $R$ is

\[\mat_R(m, n) = \fun(\llbra m \rrbra \times \llbra n \rrbra, R).\]

An $m \times n$-matrix $M$ can be explicitly written as

\[M = \begin{bmatrix} M(1, 1) & \cdots & M(1, n) \\ \vdots & \ddots & \vdots \\ M(m, 1) & \cdots & M(m, n) \end{bmatrix}.\]

Definition 2. Let $R$ be a ring and $A$, $B \in \mat_R(m,n)$ ($m$, $n \in \N^+$). The sum of $A$ and $B$ is

\[A + B \in \mat_R(m, n) : (i, j) \mapsto A(i, j) + B(i, j).\]

Definition 3. Let $R$ be a ring, $A \in \mat_R(m, n)$ and $B \in \mat_R(n, p)$ ($m$, $n \in \N^+$). The product of $A$ and $B$ is

\[AB \in \mat_R(m, p) : (i, j) \mapsto \sum_{k = 1}^n A(i, k) B(k, j).\]