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July 22, 2021 9:58 pm 30 pts
Diagonal and Trace Let A = [a,] be an n-square matrix. The diagonal or main diagonal of A Consists of the eleme- with the same subscripts, that is, a22 ag3 a The trace of A, written tr(A), is the sum of the diagonal elements. Namely, +nn tr(A) = a, t az2 +ags t+ an Theorem: Suppose A = [a] and B = [b,] are n-square matrices and k is a scalar. Then () The following theorem applies. tr(AT) = tr(A), tr(AB) = tr(BA). (ii) tr(A + B) = tr(A) + tr(B), tr(kA) = k tr(A), (iv) (i) A 2 3 2. Then and B =0 Example: Let A =| 6 2 1 5 and tr(A) = 1 - 4+7 = 4 tr(B) = 2 +3 - 4 = 1 diagonal of A = {1, 4, 7} and diagonal of B = {2, 3,-4) tr(A + B) = 3- 1 +3 5. tr(2A) = = 2 - 8+ 14 8. tr(AT) = 1 - 4+7 = tr(AB) = 5+ 0 - 35 =-30, tr(BA) = 27 24-33 =-30 Moreover, As expected from Theorem, tr(A +B) = tr(A) + tr(B), tr(AT) = tr(A), tr(2A) = 2 tr(A) Furthermore, although AB # BA, the traces are equal.
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• Alka gupta
yes, given theorem is absolutely correct and examples are also correct no single mistake in solution... 