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 wi

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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 Wi 1 (98.42 KiB) Viewed 18 times

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? choose 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 21 55 [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent (3 47 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 2, [[3,4],[5,6]] for the answer 2, (Hint: to show that H is not closed under scalar multiplication, it is sufficient to 5 find a real number and a nilpotent matrix A such that (RA)" + 0 for all positive integers n.) Bol 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 His a subspace of V His not a subspace of V