Is $(\mathbb{R}^{n\times n},+)$ resp. $(\mathbb{R}^{n\times n},\cdot)$ a group?

Question

I have to check whether $(\mathbb{R}^{n\times n},+)$ and $(\mathbb{R}^{n\times n},\cdot)$ are groups but I'm experiencing some problems. /By $\mathbb{R}^{n×n}$ are meant the matrices of size $n\times n$/

So we have the four group axioms:

(G0) : Closure $∀a,b∈G: a∘b∈G$
(G1) : Associativity $∀a,b,c∈G: a∘(b∘c)=(a∘b)∘c$
(G2) : Identity $∃e∈G:∀a∈G: e∘a=a=a∘e$
(G3) : Inverse $∀a∈G:∃b∈G: a∘b=e=b∘a$

Its obvious that the first 3 axioms are satisfied for both $(\mathbb{R}^{n\times n},+)$ and $(\mathbb{R}^{n\times n},\cdot)$

But does G3 holds since not every matrix is invertible?

Answer 1

The set of $n\times n$ matrices over the reals is not a group with multiplication. First, the zero matrix is absorbing and second not each matrix has an inverse. Additively, it is an abelian group.

Answer 2

I assume by $\mathbb{R}^{n\times n}$ you mean the matrices of size $n\times n$? If so, of course it is not a group with matrix multiplication because not every matrix is invertible. However, if you take only the invertible $n\times n$ matrices then you really get a group. (which is called the general linear group)