The trace of a square n x n matrix A = (a_ij) is the sum a_11 + a_22 + ... + a_n

Question

The trace of a square n x n matrix A = (a_ij) is the sum a_11 + a_22 + ... + a_nn 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 1 Is H a subspace of the vector space V? Does H contain the zero vector of 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 3 2 4], [5 7 6 8]. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not. enter 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 5 4 6]. 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 pans 1-3.

Explanation / Answer

If V be a vector space of 2×2 matrix

a11 a12

a21 a22 are the matrix for all aij belongs to real number

And let H be set of 2×2 matrix with trace =1

i.e , a11+a22 = 1, clearly see that H is subspace of V

1:- H contain the zero vector

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