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?

The trace of a matrix is the sum of its main diagonal elements.

1. Statement 1: For any two rectangular matrices $A$ of size $m \times n$ and $B$ of size $n \times m$, the product $AB$ is of size $m \times m$ and $BA$ is of size $n \times n$.

   The trace of $AB$ is:

   $$\operatorname{tr}(AB) = \sum_{i=1}^{m} (AB)_{ii} = \sum_{i=1}^{m} \sum_{j=1}^{n} A_{ij} B_{ji}$$

   The trace of $BA$ is:

   $$\operatorname{tr}(BA) = \sum_{j=1}^{n} (BA)_{jj} = \sum_{j=1}^{n} \sum_{i=1}^{m} B_{ji} A_{ij}$$

   Since multiplication of scalar entries is commutative, $\sum_{i=1}^{m} \sum_{j=1}^{n} A_{ij} B_{ji} = \sum_{j=1}^{n} \sum_{i=1}^{m} B_{ji} A_{ij}$, which implies $\operatorname{tr}(AB) = \operatorname{tr}(BA)$. Hence, Statement 1 is correct.

2. Statement 2: Since $C$ and $D$ are both square matrices of size $n \times n$, this is a special case of the cyclic property of the trace (with $m = n$), so $\operatorname{tr}(CD) = \operatorname{tr}(DC)$. Hence, Statement 2 is correct.

Therefore, Both Statement 1 and Statement 2 are correct, which corresponds to option (C).