The given question is about multiplying two upper triangular matrices, $A$ and $B$, which are $n \times n$ matrices. The question focuses on understanding when a term in the product of $A$ and $B$ will be zero.

The matrix multiplication is defined as: $(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$ where $A$ and $B$ are upper triangular matrices. In an upper triangular matrix, all the elements below the diagonal are zero, i.e., $a_{ik} = 0$ if $i > k$ and $b_{kj} = 0$ if $k > j$.

The question asks to identify when either $a_{ik} = 0$ or $b_{kj} = 0$, which would make the corresponding summation term zero. This occurs in the following scenarios:

• $a_{ik} = 0$ when $i > k$ because $A$ is upper triangular.
• $b_{kj} = 0$ when $k > j$ because $B$ is upper triangular.

Thus, the condition for the sum to contain zero terms is:

• $k < i$ because then $a_{ik} = 0$, and
• $k > j$ because then $b_{kj} = 0$.

The correct answer is therefore based on the combination of these two conditions, which is option B or C only.

Would you like a further breakdown of this solution, or any specific details?

Here are five questions that relate to this explanation:

1. What is an upper triangular matrix and how does it affect matrix multiplication?
2. How is matrix multiplication defined in terms of summation over an index?
3. What happens when either $a_{ik}$ or $b_{kj}$ equals zero in the multiplication?
4. What role do the indices $i$, $j$, and $k$ play in matrix multiplication?
5. Why does the structure of triangular matrices simplify determining when the result is zero?

Tip: When multiplying upper triangular matrices, focus on the indices where the nonzero entries can appear, which will always be on or above the diagonal.