Question

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?

• A. Statement 1 is correct and Statement 2 is wrong.
• B. Statement 1 is wrong and Statement 2 is correct.
• C. Both Statement 1 and Statement 2 are correct.
• D. Both Statement 1 and Statement 2 are wrong.

Answer 1

The Correct Option is C

The question concerns standard properties of the trace function of matrices. We examine each statement separately.

1. Statement 1: \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \)

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times m\) matrix. Both products \(AB\) and \(BA\) are square matrices (of orders \(m \times m\) and \(n \times n\), respectively), so their traces are well-defined.

Using the definition of trace and basic summation properties: \[ \operatorname{tr}(AB) = \sum_i (AB)_{ii} = \sum_{i,j} A_{ij} B_{ji} \] \[ \operatorname{tr}(BA) = \sum_j (BA)_{jj} = \sum_{j,i} B_{ji} A_{ij} \] Both expressions are identical sums, hence: \[ \operatorname{tr}(AB) = \operatorname{tr}(BA) \]

Therefore, Statement 1 is correct.

2. Statement 2: \( \operatorname{tr}(CD) = \operatorname{tr}(DC) \)

Here, \(C\) and \(D\) are both \(n \times n\) square matrices. Since their dimensions are compatible, the cyclic property of trace applies directly.

Using the same reasoning as above: \[ \operatorname{tr}(CD) = \sum_{i,j} C_{ij} D_{ji} = \sum_{j,i} D_{ji} C_{ij} = \operatorname{tr}(DC) \]

Thus, Statement 2 is also correct.

Conclusion: Both Statement 1 and Statement 2 are correct.