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7. defined as Let A = [a] be a square matrix of size nx n. The trace of A, denoted as Tr(A), is n =Σa i=1 Tr(A) = a11 + a22+a33 + + ann = aii. Take A, B square matrices of size nx n. Show that (i) Tr(A + B) = Tr(A) + Tr(B); (ii) Tr(AT) = Tr(A); (iii) Tr(AA) = XTr(A), where A is a nozero scalar; (iv) Tr(AB) = Tr(BA).

7. defined as Let A = [a] be a square matrix of size nx n. The trace of A, denoted as Tr(A), is n =Σa i=1 Tr(A) = a11 + a22+a33 + + ann = aii. Take A, B square matrices of size nx n. Show that (i) Tr(A + B) = Tr(A) + Tr(B); (ii) Tr(AT) = Tr(A); (iii) Tr(AA) = XTr(A), where A is a nozero scalar; (iv) Tr(AB) = Tr(BA).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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7. defined as Let A = [a] be a square matrix of size nx n. The trace of A, denoted as Tr(A), is Tr(A) = a11 + a22 + a33 +... + ann = n Σa aii. i=1 Take A, B square matrices of size n × n. Show that (i) Tr(A + B) = Tr(A) + Tr(B); (ii) Tr(AT) = Tr(A); (iii) Tr(\A) = \Tr(A), where X is a nozero scalar; (iv) Tr(AB) = Tr(BA).
Transcribed Image Text:7. defined as Let A = [a] be a square matrix of size nx n. The trace of A, denoted as Tr(A), is Tr(A) = a11 + a22 + a33 +... + ann = n Σa aii. i=1 Take A, B square matrices of size n × n. Show that (i) Tr(A + B) = Tr(A) + Tr(B); (ii) Tr(AT) = Tr(A); (iii) Tr(\A) = \Tr(A), where X is a nozero scalar; (iv) Tr(AB) = Tr(BA).
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