📘 Question
If \(A\) and \(B\) are symmetric matrices, then \(ABA\) is:
(a) symmetric matrix
(b) skew-symmetric matrix
(c) diagonal matrix
(d) scalar matrix
✏️ Step-by-Step Solution
Step 1: Use symmetry condition
\[
A^T = A,\quad B^T = B
\]
Step 2: Take transpose of \(ABA\)
\[
(ABA)^T = A^T B^T A^T
\]
Step 3: Substitute
\[
= A B A
\]
Step 4: Conclusion
Since:
\[
(ABA)^T = ABA
\]
Therefore, \(ABA\) is a symmetric matrix.
✅ Final Answer
\[
\boxed{(a)\; \text{symmetric matrix}}
\]
💡 Key Concept
Transpose of a product reverses order:
\[
(ABC)^T = C^T B^T A^T
\]
Use symmetry condition to simplify.