Answer:
Remember, a set [tex]S\subset V[/tex]is a subspace of the vector space V if:
[tex]1. 0\in S\\2. a+b\in S,\text{ for all } a,b\in S\\3. \lambda a\in S, \text{for } \lambda \text{ a scalar}[/tex].
1. Observe that
[tex]0\times 0=0^2=0[/tex], where [tex]0[/tex] is the null matrix [tex]2\times 2[/tex].
Then [tex]0 \in H[/tex]
2. Consider two matrices A and B such that [tex]AB\neq 0[/tex] in H
Observe that [tex](A+B)^2=A^2+2AB+B^2\\=A+2AB+B\neq A+B[/tex].
Then, [tex]A+B\nsubseteq H[/tex]
Then H is not a subspace of V
