Prove that if A and B are n x n matrices, then tr(AB) = tr(BA).

Chardonnay Felix 2020-11-12 Answered
Prove that if A and B are n x n matrices, then tr(AB) = tr(BA).
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Expert Answer

hosentak
Answered 2020-11-13 Author has 100 answers
Step 1
Let A and B are n×n matrices.
Consider the AB=[cij]n×n with , cij=k=1naikbkj
Consider the BA=[dij]n×n with , dij=k=1nbikakj
=s=1nbisasj
Step 2
Use the continuation for the above expression,
tr(AB)=i=1ncij
i=1n(k=1naikbki)
Step 3
Evaluate the above expression by interchanging the order of summation,
tr(AB)=i=1n(s=1naksbks)
k=1ndkk
i=1ndii
=tr(BA)
Hence proved tr(AB)=tr(BA).
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Jeffrey Jordon
Answered 2022-01-24 Author has 2047 answers

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