Which of the following sets of $n \times n$ matrices with real entries is a vector space over $\mathbb R$ if the vector addition and scalar multiplication are as usual?
A) the set of invertible matrices,
B) the set of non-invertible matrices,
C) the set of matrices with a zero diagonal,
D) the set of non-zero matrices,
E) none of the above.
I have the answer. It is C. I just do not understand why. I thought it must be A because of the axiom that says there must be an inverse of each element. Why the zero diagonal?