Please solve and answer in the same format as the question as it makes it easier to follow. Box answers and do not type it out and I will give you a thumbs up rating if answered correctly.
Chapter 4.1 Question 2
**Part two and three are wrong please tell me what to put in the boxes**
Extracted text: You have answered 2 out of 4 parts correctly. A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty? H is nonempty v O 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 21 [5 6] 1E . (Hint: to show that H is not closed under addition, it is sufficient to find two 7 8 answer nilpotent matrices A and B such that (A+ B)" + 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 R and a matrix in H whose product is not in H, using a comma separated list and syntax such as [3 47 2, [[3,4], [5,6]] for the answer 2, 5 6 (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)" + 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. H is not a subspace of V v O