Its been a while but if I remember correctly you have to prove the following:
1. For any given matrix (F is a scalar field, usually ) whose determinant , and for every scalar , the matrix is also element of the space and has the same property as the matrix A: .
This one is easy to prove. First, it is obvious that: the matrix , no doubt about that. Just recall the manner in which matrices are multiplied by scalars. Now one has to see if the matrix has the same property as the matrix A. If one can prove that then we proved that subset is closed under multiplication by scalars. And this is easy too:
( because you extract one per row of the determinant).
2. One still has to prove that for any two matrices A and B, both having zero determinant values, matrix A+B will also have a zero determinant value. Now, I hope someone will correct me if I'm wrong but I think this is not the case. A counterexample comes to mind:
Matrices and both have zero determinants, but the matrix does not. .
So it turns out that it is not a subspace closed under matrix/vector addition.
I hope I didn't make a mistake somewhere. As I said, its been a while...
Someone else, please.