A square matrix $A$ is nilpotent if $A^n = \boldsymbol{0}$ for some positive integer $n$.

Let $V$ be the vector space of all $2 \times 2$ matrices with real entries. Let $H$ be the set of all $2 \times 2$ nilpotent matrices with real entries. Is $H$ a subspace of the vector space $V$?

1. Is $H$ nonempty?
   choose H is empty H is nonempty
   
   
2. Is $H$ closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in $H$ whose sum is not in $H$, using a comma separated list and syntax such as $\verb+[[1,2],[3,4]], [[5,6],[7,8]]+$ for the answer $\left[\begin{array}{cc} 1 &2\cr 3 &4 \end{array}\right],\left[\begin{array}{cc} 5 &6\cr 7 &8 \end{array}\right]$. (Hint: to show that $H$ is not closed under addition, it is sufficient to find two nilpotent matrices $A$ and $B$ such that $(A+B)^n \ne \boldsymbol{0}$ for all positive integers $n$.)
   
   
   
3. Is $H$ closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in $\mathbb{R}$ and a matrix in $H$ whose product is not in $H$, using a comma separated list and syntax such as $\verb+2, [[3,4],[5,6]]+$ for the answer $2, \left[\begin{array}{cc} 3 &4\cr 5 &6 \end{array}\right]$. (Hint: to show that $H$ is not closed under scalar multiplication, it is sufficient to find a real number $r$ and a nilpotent matrix $A$ such that $(rA)^n \ne \boldsymbol{0}$ for all positive integers $n$.)
   
   
   
4. Is $H$ a subspace of the vector space $V$? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.
   choose H is a subspace of V H is not a subspace of V