📘 Question
If \(A\) is of order \(m \times n\) and \(B\) is such that:
\[
AB^T \quad \text{and} \quad B^T A
\]
are both defined, find the order of matrix \(B\).
(a) \(m \times n\)
(b) \(n \times n\)
(c) \(n \times m\)
(d) \(m \times n\)
✏️ Step-by-Step Solution
Step 1: Let order of \(B\)
\[
B = p \times q \Rightarrow B^T = q \times p
\]
—
Step 2: Condition for \(AB^T\)
\[
A(m \times n) \cdot B^T(q \times p)
\]
For multiplication:
\[
n = q
\]
—
Step 3: Condition for \(B^T A\)
\[
B^T(q \times p) \cdot A(m \times n)
\]
For multiplication:
\[
p = m
\]
—
Step 4: Final order of \(B\)
\[
B = p \times q = m \times n
\]
—
✅ Final Answer
\[
\boxed{(a)\; m \times n}
\]
—
💡 Key Concept
Use transpose dimensions carefully:
- \(B^T\) flips rows and columns
- Match inner dimensions for multiplication