If a matrix A is both symmetric and skew-symmetric, then :

The correct answer is Option (2) → A is a zero matrix

A is symmetric ⇒ $A=A^T$

A is skew-symmetric ⇒ $A^T=A$

if both hold,

$A=A^T=-A$

$∴A=-A⇒2A=0⇒A=0$