A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? 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 [[1,2], [3,4]], [[5,6],[7,8]] for the answer BJ6 [1 2] [5 6] 8 (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)" ‡ 0 for all positive integers n.) Closed 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax [3 such as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed 5 6 under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA) #0 for all positive integers n.) Closed 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. H is not a subspace of V

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