Symmetric matrices are square.

Fact. A symmetric matrix $A$ is always square (i.e., $n \times n$).

Reason: For $A = A^T$ to hold, $A$ must have the same number of rows and columns. If $A$ is $m \times n$, then $A^T$ is $n \times m$, so equality requires $m = n$.

This distinguishes symmetric matrices from rectangular matrices, for which the notion of symmetry (equality with transpose) is not defined.