Discussion
Consider the following two statements with respect to the matrices \(A_{m \times n}\), \(B_{n \times m}\), \(C_{n \times n}\) and \(D_{n \times n}\).
Statement 1: tr(AB) = tr(BA)
Statement 2: tr(CD) = tr(DC)
where tr() represents the trace of a matrix. Which one of the following holds?
Answer: C
Both statements are correct.
For \(A\) of size \(m \times n\) and \(B\) of size \(n \times m\), \(AB\) is \(m \times m\) and \(BA\) is \(n \times n\), so both traces exist. Each is the same double sum:
\[ \begin{aligned} tr(AB) &= \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} b_{ji} \\ tr(BA) &= \sum_{j=1}^{n} \sum_{i=1}^{m} b_{ji} a_{ij} \end{aligned} \]
The terms are identical, only summed in a different order, so \(tr(AB) = tr(BA)\) even though the two products have different sizes. Statement 2 is the special case \(m = n\), so it is correct too. The options calling either statement wrong confuse \(tr(AB) = tr(BA)\) with \(AB = BA\), which does fail in general.
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