Subgroups of specific matrices I believe $\mathbb{M}_3(\mathbb{R})$ is the set of all 3x3 matrices,  $GL_3(\mathbb{R})$ is a subset of $\mathbb{M}_3(\mathbb{R})$ 
and $GL_3(\mathbb{R})$ is not a subgroup of $\mathbb{M}_3(\mathbb{R})$. 
I also know $SL_3(\mathbb{R})$ is a subgroup of $GL_3(\mathbb{R})$. But I was wondering if that means $SL_3(\mathbb{R})$ is also not a subgroup of  $\mathbb{M}_3(\mathbb{R})$. Or can $SL_3(\mathbb{R})$ be a subgroup of $\mathbb{M}_3(\mathbb{R})$?
 A: As the comments point out, the question as written is ill-posed: to define a group you need to define both the set of elements of the group and the group operation. For matrices in particular, there are two "natural" candidates for the group operation: addition and matrix multiplication.
The set of all $3\times 3$ matrices is a group under addition, but not under multiplication (because the zero matrix, along with all other singular matrices, does not have an inverse.)
$[GL_3(\mathbb{R}),+]$ is not a subgroup of $[M_3(\mathbb{R}),+]$ since $GL_3(\mathbb{R})$ does not contain the zero matrix (among other issues).
As you say, $[SL_3(\mathbb{R},\times]$ is a subgroup of $[GL_3(\mathbb{R}),\times]$: it is easy to check that the set of matrices with determinant 1 contains the identity, and is closed under multiplication and taking matrix inverse. $[SL_3(\mathbb{R}),\times]$ is not a subgroup of $[M_3(\mathbb{R}),\times]$ since the latter is not a group, and $[SL_3(\mathbb{R}),+]$ is not a subgroup of $[M_3(\mathbb{R}),+]$ since it does not contain the zero matrix.
