matrices vector spaces

Question

Consider the vector space of $3\times3$ matrices with real coefficients.

Let $W$ denote the subset of matrices with determinant $0$.

Decide whether $W$ is a subspace or not.

Answer 1

It is sufficient to check whether the set is closed under scalar multiplication and vector addition.

Thus, you must explore whether the sum of two zero-determinant matrices also has a zero determinant, and whether the scalar multiple of a zero-determinant matrix is also a zero-determinant matrix.

If you find it to be so, write a proof. If you find it to not be so, offer a counter example.

Answer 2

The answer is that $W$ is not a subspace. In order to prove that this is the case, try to find two matrices with determinant $0$ that sum to a matrix with non-zero determinant.