If A and B are symmetric matrices of same order, then which of the fo…

The correct answer is Option (4) → (B) and (C) only

A and B are symmetric, so \( A^T=A \) and \( B^T=B \). By the reversal law, \( (XY)^T=Y^TX^T \); in (C) and (D) the inverses are assumed to exist.

For (A), let

    \( \displaystyle P=AB^T+B^TA=AB+BA \)

    \( \displaystyle P^T=(AB)^T+(BA)^T \)

    \( \displaystyle =BA+AB=P \)

For (B), let

    \( \displaystyle Q=AB^T-BA^T=AB-BA \)

    \( \displaystyle Q^T=(AB)^T-(BA)^T \)

    \( \displaystyle =BA-AB=-Q \)

For (C):

    \( \displaystyle (AB)^T(A^T)^{-1}=B^TA^T(A^T)^{-1} \)

    \( \displaystyle =B^T=B \)

For (D):

    \( \displaystyle (AB)^{-1}(BA)^T=B^{-1}A^{-1}A^TB^T \)

    \( \displaystyle =B^{-1}A^{-1}AB \)

    \( \displaystyle =B^{-1}B=I \)

So AB^T + B^TA is symmetric, AB^T − BA^T is skew-symmetric, (AB)^T(A^T)⁻¹ = B and (AB)⁻¹(BA)^T = I.

Checking the statements

(A) AB^T + B^TA is a skew-symmetric matrix. — Incorrect. \( P^T=P \), so it is symmetric (skew-symmetric only if it is the zero matrix).

(B) AB^T - BA^T is a skew-symmetric matrix — Correct. \( Q^T=-Q \).

(C) (AB)^T (A^T)^-1 = B — Correct. \( (A^T)^{-1} \) cancels \( A^T \), leaving \( B^T=B \).

(D) (AB)^-1 (BA)^T = A — Incorrect. The product is I, not A.