The trace of a square n x n matrix A = (a;j) is the sum a11 + a22 + - + ann of the entries on its main diagonal. ... 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 matrices with real entries that have trace 0. Is H a subspace of the vector space V? 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 [1 2] [5 6] 3 4 (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices A and B such that A +B has nonzero trace.) 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, 3 4 - (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)

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