Question 20

For any matrices $A_{m \times n}$ and $B_{n \times m}$:

• The product $AB$ is an $m \times m$ matrix with diagonal entries $(AB)_{ii} = \sum_{k=1}^n A_{ik} B_{ki}$.
• The product $BA$ is an $n \times n$ matrix with diagonal entries $(BA)_{jj} = \sum_{l=1}^m B_{jl} A_{lj}$.

Computing the traces:

Hence, Statement 1 holds for all rectangular matrices $A$ and $B$.

Statement 2 is a direct instance of the same identity where $C$ and $D$ are square matrices of size $n \times n$. Therefore, $\text{tr}(CD) = \text{tr}(DC)$ is also correct.

Both Statement 1 and Statement 2 are correct.