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.)

Let A B denotes the tensor product of matrices Amxn and Bpxq defined by the block matrix A B = a12B 922B a11B a21 B : : ami B am2 B : ain B a2n B : amn B mpxnq (a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr. Prove that (AB)(C > D) = ACⒸ BD (b) Let Amxm and B₁xn be nonsingular matrices. - Prove that (AB) is nonsingular. - Find the inverse matrix of (AB) (c) Prove that the following equality applies for any Amxm and Bnxn square matrices trace(AB) = trace(A) * trace(B)

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 × 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 [[1,2], [3,4]], [[5,6],[7,8]] for the answer [1 2] [56] 3.6 . (Hint: to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A 3 4 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 [34] 2, 3] (Hint: to show that is not closed under scalar multiplication, it is sufficient to find a real…

Please solve this question with a good explanation. It is Linear Algebra question.

Let V be a vector space of dimension n, and let B and C be bases for V. Then the product PB- cPc-B is equal to the n × n identity matrix. True False

Let V be the vector space of all (2 × 2) matrices. Determine whether the followingmatrices A, B, and C are linearly independent or dependent?A =(1 23 1 ), B =(3 −12 2 ), C =(1 −5−4 0 )

Find a basis of the subspace of R5 spanned by the following vectors: 4 1 -1 8 6 6 -2 2 12 16 -1 -2 2 -2 2 5 3 -2 10 4 5 3 -2 10 4