If A is a skew symmetric matrix, then \( A^{2} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Symmetric matrix
A is skew symmetric, so \( A^{T}=-A \). Using \( (PQ)^{T}=Q^{T}P^{T} \):
\( (A^{2})^{T}=(AA)^{T}=A^{T}A^{T} \)
\( =(-A)(-A)=A^{2} \)
Since \( (A^{2})^{T}=A^{2} \), \( A^{2} \) is a symmetric matrix.
Why the other options are incorrect
Option (1) Skew symmetric matrix — Incorrect. Skew symmetric would need \( (A^{2})^{T}=-A^{2} \), but the two minus signs cancel and give \( +A^{2} \).
Option (2) Null matrix — Incorrect. \( A^{2} \) need not be zero: for \( A=\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix} \), \( A^{2}=\begin{bmatrix} -1 & 0\\ 0 & -1\end{bmatrix} \).
Option (4) Row matrix — Incorrect. A skew symmetric matrix is square, so \( A^{2} \) is a square matrix, not a row matrix.
Tip: Even powers of a skew symmetric matrix are symmetric; odd powers are skew symmetric.
NCERT Exemplar:
Matrices
