If A and B are two matrices of order 2 × 2 such that A is a symmetric matrix and B is a skew-symmetric matrix, then:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $A^2$ is also a symmetric matrix
For any symmetric matrix $A$, $A^T=A$.
$(A^2)^T=(AA)^T=A^TA^T=AA=A^2$, so $A^2$ is symmetric ✅
For any skew-symmetric matrix $B$, $B^T=-B$.
$(B^2)^T=(BB)^T=B^TB^T=(-B)(-B)=BB=B^2$, so $B^2$ is symmetric, not skew-symmetric ❌
$A+B$ cannot be an identity matrix in general ❌
$A-B$ cannot be a null matrix in general ❌
Correct answer: $A^2$ is also a symmetric matrix.