Answer
True
Work Step by Step
$A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33} \end{bmatrix}$ with characteristic polynomial $p(\lambda)=\lambda^3+\lambda \rightarrow \lambda^3+\lambda =0\\
\rightarrow \lambda(\lambda^2+1)=0\\
\rightarrow \lambda_1=0\\
\lambda_2=i\\
\lambda_3=-i$
We can see that $A$ has two complex eigenvalues. According to theorem 7.5.4, the matrix $A$ can not be symmetric.
Hence, the statement is true.
