If A = [aij] is an n × n matrix, then the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A, Tr(A) = .Show each of the following:
(a) Tr(cA) − cTr(A), where c is areal number
(b) Tr(A + B) − Tr(A) + Tr(5)
(c) Tr(AB) = Tr(BA)
(d) Tr(AT) = Tr (A)
(e) Tr(ATA) ≥ 0
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