📘 Question
If \(A\) and \(B\) are matrices of suitable order, then:
\[
AB^T – B^T A
\]
is a:
(a) skew-symmetric matrix
(b) null matrix
(c) unit matrix
(d) symmetric matrix
✏️ Step-by-Step Solution
Step 1: Take transpose
\[
(AB^T – B^T A)^T = (AB^T)^T – (B^T A)^T
\]
Step 2: Use transpose property
\[
(AB^T)^T = B A^T,\quad (B^T A)^T = A^T B
\]
\[
(AB^T – B^T A)^T = BA^T – A^T B
\]
Step 3: Rearrange
\[
= -(AB^T – B^T A)
\]
Step 4: Conclusion
Since:
\[
M^T = -M
\]
Matrix is skew-symmetric.
✅ Final Answer
\[
\boxed{(a)\; \text{skew-symmetric matrix}}
\]
💡 Key Concept
A matrix is skew-symmetric if:
\[
A^T = -A
\]
Use transpose of product:
\[
(AB)^T = B^T A^T
\]