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For an n×n matrix, A, the trace of A is defined as the sum of the...

For an n×n matrix, A, the trace of A is defined as the sum of the entries on the main diagonal. That is, tr(A)=a11+a22+?+ann.

(a) Prove that for any matrices A and B having the same size, tr(A+B)=tr(A)+tr(B) and for any scalar c, tr(cA)=ctr(A)

(b) Prove tr(A)=tr(AT) for all square matrices A.

(c) Prove that for any matrices A and B having the same size, tr(AB)=tr(BA).

(d) Using (c), prove that if A and B are similar tr(A)=tr(B).

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