If A is a skew-symmetric matrix then, the matrix B^TAB is:-

The correct answer is Option (4) → Skew-symmetric matrix

A is skew-symmetric, so \( A^T=-A \). The transpose of \( B^TAB \) follows from the reversal law \( (PQR)^T=R^TQ^TP^T \):

    \( \displaystyle (B^TAB)^T=B^TA^T(B^T)^T \)

    \( \displaystyle =B^T(-A)B \)

    \( \displaystyle =-(B^TAB) \)

So \( B^TAB \) is a skew-symmetric matrix.

Tip: The result "\( B^TAB \) keeps the type of A" works both ways: if A is symmetric, \( B^TAB \) is symmetric; if A is skew-symmetric, \( B^TAB \) is skew-symmetric.

NCERT (2026-27 edition):
Matrices