Which of the following statements is (are) true?

(A) $B^T AB$ is a skew-symmetric matrix if A is a symmetric matrix

(B) $B^T AB$ is a symmetric matrix if A is a symmetric matrix

(C) $B^T AB$ is a symmetric matrix if A is a skew-symmetric matrix

(D) $B^T AB$ is a skew-symmetric matrix if B is a skew-symmetric matrix

(E) $B^T AB$ is a symmetric matrix if B is a symmetric matrix

Choose the correct answer from the options given below:

A matrix is symmetric if $M^T = M$ and skew-symmetric if $M^T = -M$.

When taking transpose of a product: $(AB)^T = B^T A^T$

---

Testing Statement (A): $B^T AB$ is skew-symmetric if A is symmetric

Take the transpose of $B^T AB$:

$(B^T AB)^T = B^T A^T (B^T)^T$

$(B^T AB)^T = B^T A^T B$

For A symmetric, $A^T = A$:

$(B^T AB)^T = B^T A B$

$(B^T AB)^T = B^T AB$

For skew-symmetric, we need $(B^T AB)^T = -(B^T AB)$, but the result shows $(B^T AB)^T = B^T AB$.

Statement (A) is false.

---

Testing Statement (B): $B^T AB$ is symmetric if A is symmetric

Take the transpose:

$(B^T AB)^T = B^T A^T B$

For A symmetric, $A^T = A$:

$(B^T AB)^T = B^T A B$

$(B^T AB)^T = B^T AB$

This satisfies the definition of symmetric.

Statement (B) is true.

---

Testing Statement (C): $B^T AB$ is symmetric if A is skew-symmetric

Take the transpose:

$(B^T AB)^T = B^T A^T B$

For A skew-symmetric, $A^T = -A$:

$(B^T AB)^T = B^T(-A)B$

$(B^T AB)^T = -B^T AB$

For symmetric, we need $(B^T AB)^T = B^T AB$, but the result is $(B^T AB)^T = -B^T AB$.

Statement (C) is false.

---

Testing Statement (D): $B^T AB$ is skew-symmetric if B is skew-symmetric

This requires specific conditions on A as well and is not generally true.

Statement (D) is false.

---

Testing Statement (E): $B^T AB$ is symmetric if B is symmetric

For B symmetric, $B^T = B$:

$(B^T AB)^T = B A^T B$

For symmetry, we need $BA^T B = BAB$, which requires $A^T = A$. However, A being symmetric is not given.

Statement (E) is false.

---

Only statement (B) is true: When A is symmetric, $B^T AB$ is always symmetric, regardless of B.

The answer is Option 3: (B) only