1. **Problem:** Find the order of $(AB)^t$ given $A = [a_{ij}]_{m \times n}$ and $B = [b_{ij}]_{n \times r}$. 2. **Recall:** The order (dimensions) of matrix $A$ is $m \times n$ and matrix $B$ is $n \times r$. 3. **Matrix multiplication rule:** The product $AB$ is defined and has order $m \times r$ because the inner dimensions $n$ match. 4. **Transpose rule:** The transpose of a matrix $M$ of order $p \times q$ is $M^t$ of order $q \times p$. 5. **Apply to $(AB)^t$:** Since $AB$ is $m \times r$, then $(AB)^t$ is $r \times m$. 6. **Check options:** - a. $m \times r$ (incorrect, this is order of $AB$ not its transpose) - b. $n \times m$ (incorrect) - c. $m \times n$ (incorrect) - d. None of these (correct, since $(AB)^t$ is $r \times m$ which is not listed) **Final answer:** $(AB)^t$ has order $r \times m$, so the correct choice is d. None of these.