Let A be a non-singular matrix of order  three such that  $A=adj(A+A^T)$ then which of the following statement(s) is(are) TRUE?

Apply transpose to the given equation, we get  $A=A^T$
A is symmetric matrix
$$\begin{gathered} A=Adj\left(2A\right) \\ \left|A\right|=\frac{1}{64} \\ 2A^2=2Aadj\left(2A\right)=\left|2A\right|I=8\left|A\right|I=\frac{I}{8} \\ {A}^{-1}=16A \\ Adj\left(A^{-1}\right)=Adj\left(16A\right)=64Adj\left(2A\right)=64A \end{gathered}$$