# 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 $AB={\left[{c}_{ij}\right]}_{n×n}$ with , ${c}_{ij}=\sum _{k=1}^{n}{a}_{ik}{b}_{kj}$
Consider the $BA={\left[{d}_{ij}\right]}_{n×n}$ with , ${d}_{ij}=\sum _{k=1}^{n}{b}_{ik}{a}_{kj}$
$=\sum _{s=1}^{n}{b}_{is}{a}_{sj}$
Step 2
Use the continuation for the above expression,
$tr\left(AB\right)=\sum _{i=1}^{n}{c}_{ij}$
$\sum _{i=1}^{n}\left(\sum _{k=1}^{n}{a}_{ik}{b}_{ki}\right)$
Step 3
Evaluate the above expression by interchanging the order of summation,
$tr\left(AB\right)=\sum _{i=1}^{n}\left(\sum _{s=1}^{n}{a}_{ks}{b}_{ks}\right)$
$\sum _{k=1}^{n}{d}_{kk}$
$\sum _{i=1}^{n}{d}_{ii}$
$=tr\left(BA\right)$
Hence proved tr(AB)=tr(BA).
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Jeffrey Jordon
Answered 2022-01-24 Author has 2047 answers

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