# Is matrix multiplication really a group operator?

A group has an operation that can be performed over ANY two elements in a set. Given that an $n \times m$ matrix can only be multiplied by an $m \times o$ matrix, doesn't that mean that matrix multiplication can't be a group operator except for sets of commonly sized square matrices?

I ask this because it's called linear algebra yet this aspect seems inconsistent with groups.

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I am not sure what the term linear algebra has to do with the question (why should the set of matrices be a group due to that name?) – Tobias Kildetoft Nov 26 '13 at 18:57
There is a lot more to algebra than just groups. – Dustan Levenstein Nov 26 '13 at 18:57
I guess I've been confused then, isn't an algebra also a ring, which includes two groups? – Hans Nov 26 '13 at 19:04
No, "an" algebra is indeed in particular a ring. But a ring only includes one group. And the term algebra refers to an area of mathematics, not a mathematical object, when one says "linear algebra" (and probably the term is more often used for the area than for the object actually). – Tobias Kildetoft Nov 26 '13 at 19:08
Thanks for clarifying, that's what tripped me up. – Hans Nov 26 '13 at 19:15

Usually, when we talk about groups of matrices, we talk about square matrices, as only then can we be guaranteed that $AB$ and $BA$ are valid operations. In fact, if we refer to matrices as a group, we talk about invertible matrices, and we usually write it as $GL_n(\mathbb{F})$, or the general linear group of $n \times n$ matrices over some field $\mathbb{F}$.

The set of all matrices does not form a group under matrix multiplication or addition.

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"All matrices" does not even describe a set. – Tobias Kildetoft Nov 26 '13 at 18:59
I couldn't think of the informal word to describe "all matrices" >.< – Emily Nov 26 '13 at 19:00
It does over a fixed field $\mathbb F$. – Dustan Levenstein Nov 26 '13 at 19:00
@Arkamis A bunch of something which is not necessarily a set is usually referred to as a family. – Vedran Šego Nov 26 '13 at 19:19
@VedranŠego ;) Thanks! – Emily Nov 26 '13 at 20:32

When people talk about matrix multiplication being a group, they usually refer to the set of invertible $n\times n$ matrices over a field $F$.

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