If A and B are symmetric matrices of same order, then AB – BA is a
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Skew-symmetric matrix
Given \( \mathrm{A}^{\mathrm{T}}=\mathrm{A} \) and \( \mathrm{B}^{\mathrm{T}}=\mathrm{B} \). By the reversal law \( (\mathrm{XY})^{\mathrm{T}}=\mathrm{Y}^{\mathrm{T}}\mathrm{X}^{\mathrm{T}} \):
\( \displaystyle (\mathrm{AB}-\mathrm{BA})^{\mathrm{T}}=(\mathrm{AB})^{\mathrm{T}}-(\mathrm{BA})^{\mathrm{T}} \)
\( \displaystyle =\mathrm{B}^{\mathrm{T}}\mathrm{A}^{\mathrm{T}}-\mathrm{A}^{\mathrm{T}}\mathrm{B}^{\mathrm{T}} \)
\( \displaystyle =\mathrm{BA}-\mathrm{AB} \)
\( \displaystyle =-(\mathrm{AB}-\mathrm{BA}) \)
A matrix whose transpose is its negative is skew-symmetric, so AB − BA is a skew-symmetric matrix.
Tip: AB − BA becomes the zero matrix only in the special case when A and B commute; for symmetric A and B it is always skew-symmetric, while AB + BA is always symmetric.
NCERT (2026-27 edition):
Symmetric and Skew Symmetric Matrices
