📘 Question
If \(A\) is a \(3 \times 4\) matrix and \(B\) is such that both:
\[
A^T B \quad \text{and} \quad BA^T
\]
are defined, find the order of matrix \(B\).
(a) \(3 \times 4\)
(b) \(3 \times 3\)
(c) \(4 \times 4\)
(d) \(4 \times 3\)
✏️ Step-by-Step Solution
Step 1: Find \(A^T\)
\[
A = 3 \times 4 \Rightarrow A^T = 4 \times 3
\]
Step 2: Condition for \(A^T B\)
For multiplication:
\[
(4 \times 3) \cdot B \Rightarrow B \text{ must be } 3 \times n
\]
Step 3: Condition for \(BA^T\)
\[ B \cdot (4 \times 3) \Rightarrow B \text{ must be } m \times 4 \]
Step 4: Combine both conditions
So, \(B\) must be:
\[
3 \times 4
\]
✅ Final Answer
\[
\boxed{(a)\; 3 \times 4}
\]
💡 Key Concept
Matrix multiplication is defined only when inner dimensions match. Use both conditions to determine unknown matrix order.