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

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

Question
Matrices
asked 2020-11-12
Prove that if A and B are n x n matrices, then tr(AB) = tr(BA).

Answers (1)

2020-11-13
Step 1
Let A and B are n×n matrices.
Consider the \(AB=\left[c_{ij}\right]_{n \times n}\) with , \(c_{ij}=\sum_{k=1}^n a_{ik}b_{kj}\)
Consider the \(BA=\left[d_{ij}\right]_{n \times n}\) 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,
\(tr(AB)=\sum_{i=1}^n c_{ij}\)
\(\sum_{i=1}^n (\sum_{k=1}^n a_{ik}b_{ki})\)
Step 3
Evaluate the above expression by interchanging the order of summation,
\(tr(AB)=\sum_{i=1}^n (\sum_{s=1}^n a_{ks}b_{ks})\)
\(\sum_{k=1}^n d_{kk}\)
\(\sum_{i=1}^n d_{ii}\)
\(=tr(BA)\)
Hence proved tr(AB)=tr(BA).
0

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