Does the vector space of all $3\times3$ matrices include $3\times3$ matrices with $0$ as their determinants?

Question

I'm still confused as to how matrices, vector spaces, and subspaces relate to each other. If a vector space $V$ contains all $3\times3$ matrices wouldn't that also mean that matrices whose determinants are $0$ also be included?

Answer 1

Yes of course the vector space of 3-by-3 (or n-by-n) matrices includes all the matrices and thus also those with determinant equal to zero.

What is not true is that the set of all matrices with determinant equal to zero is a subspace, can you see why?

Answer 2

The space of $3×3$ matrices over a field $\mathbb F$ is just $\mathbb F^9$. This includes all the matrices with determinant zero... The reason is that we get all matrices period, including of any given determinant. That is the same as all $9$-tuples of elements of $\mathbb F$. For, there are $9$ entries in a $3×3$ matrix; and each entry can be any element of $\mathbb F$...