Define the smallest subspace of $3\times 3$ matrix vector space that contains the set of all invertible matricies.

Question

Define the smallest subspace of $3\times 3$ matrix vector space that contains the set of all invertible matrices.

My attempt:

$A$ is invertible if and only if its columns form a basis in $F^3$.

The only subspace I can think of is the vector space of all $3\times 3$ matrices, which I cannot prove is the smallest.

Note: We are not allowed to use determinant.

Answer 1

I have a topological proof in mind, even if this is probably a more elementary proof that is expected here.

• The set of $n \times n$ invertible matrices is dense in the vector space of $n \times n$ matrices.
• Moreover, a finite dimensional vector subspace is always a closed set

Therefore, if there is a subspace that contains all invertible matrices, since it is closed it contains all matrices, so it is the vector space of all $n \times n$ matrices.