Is the set of all invertible $n \times n$ matrices a vector space? I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication.  I found that is true, but I'm not sure how to prove it. 

My problem here is that this statement is too "broad", i.e. I cannot create a matrix with arbitrary values a,b,c,d,(...) considering that I don't know the size of the matrix.

My idea was to prove in a first time that the set of all invertible 2 x 2 matrices of real number is a vector space and then to show that this property could be extended to bigger matrices. However I don't know how to do it and it's precisely here that I need a little help ; how can I show that we can extend our statement?

Is it enough to say that the matrix's size doesn't affect in any way properties we need to check? 
Is my overall strategy wrong?
Any form of help would be very appreciated on this dubious answer.
The set of all invertible $n\times n$ matrices of real numbers is NOT a vector space.

Let for example $I$, the unit matrix is invertible and so is $-I$. But their sum $I+(-I)=0$ is definitely not invertible!
The set of all invertible $n \times n$ matrices is not a vector space with respect to the typical matrix addition and scalar multiplication operations and the typical matrix zero.

However,

The special orthogonal group (rotation matrices) is a vector space if you use matrix multiplication for the addition operator and the identity matrix as the zero matrix.

You also have to make scalar multiplication exponentials.

$ s \otimes A = A^s$

You can also change the zero matrix to be a matrix $C$ with:

$A\oplus B = A B C^{-1}$

and

$ s \otimes A = A^s C^{-s}$.

Then,

$ A \oplus C = A$.

And invertible $n \times n$ matrices are just rotation matrices with a scaling part and so are also vector spaces.