The trace of a n à n matrix A MnÃn. is defined to be the sum of

The trace of a n × n matrix A ˆˆ Mn×n. is defined to be the sum of its diagonal entries: tr A =
a11 + a22 +ˆ™ ˆ™ ˆ™ + ann.
(a) Compute the trace of

(b) Prove that tr(A + B) = tr A + tr B.
(c) Prove that tr(A B) = tr(BA).
(d) Prove that the commutator matrix C = A B - B A has zero trace: tr C = 0.
(e) Is pan (c) valid if A has size m × n and B has size n × m?

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