Matrix ring

Description

Let $R$ be a ring and let $n \ge 1$ be an integer. The matrix ring $M_n(R)$ is the set of $n \times n$ matrices over $R$, with matrix addition and matrix multiplication. It is a ring, with the zero matrix as zero and the identity matrix $I_n$ as unit. If $R$ is commutative, then $M_n(R)$ is also an algebra over $R$, with scalar multiplication.

Remarks

For $n \ge 2$ and $R \ne 0$, the ring $M_n(R)$ is not commutative and has nonzero elements whose product is zero. The units of $M_n(R)$ are the invertible matrices, and they form the general linear group. For a field $F$, the ring $M_n(F)$ is isomorphic to the ring of endomorphisms of $F^n$.

Explanations

L1 human

Square matrices of one fixed size can be added and multiplied, and the result is again a square matrix of that size. They follow most of the rules of numbers. The main differences are that $AB$ can differ from $BA$, and that $AB$ can be zero when $A$ and $B$ are not.