Problem 4. If A = [aij] is an nxn matrix, then the trace of A, Tr(A), is defined as the sum of all the elements on the main diagonal of A, i.e., Tr(A) = > a;;. Show each of the following: aji. i=1...

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Extracted text: Problem 4. If A = [aij] is an nxn matrix, then the trace of A, Tr(A), is defined as the sum of all the elements on the main diagonal of A, i.e., Tr(A) = > a;;. Show each of the following: aji. i=1 (iii) Tr(AB) = Tr(BA) (iv) Tr(A") = Tr(A) (υ) Tr(ATA) >0