For any square matrix A, A - \( A^T \) is always
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A skew-symmetric matrix
The transpose of \( A-A^T \) is its negative:
\( \displaystyle (A-A^T)^T=A^T-A \)
\( \displaystyle =-(A-A^T) \)
A matrix whose transpose is its negative is skew-symmetric, so \( A-A^T \) is always skew-symmetric.
Why the other options are incorrect
Option (1) A null matrix — Incorrect. \( A-A^T \) is the null matrix only when A is symmetric, not for every A.
Option (2) A symmetric matrix — Incorrect. It is \( A+A^T \) that is always symmetric.
Option (4) An identity matrix — Incorrect. Every diagonal entry of \( A-A^T \) is \( a_{ii}-a_{ii}=0 \), so it can never be the identity matrix.
CBSE Applied Mathematics (Class XII):
Symmetric and Skew Symmetric Matrices
