for any symmetric matrices $A$ and $B$ and any scalars $a$ and $b$, $(aA+bB)^t=aA^t$+$bB^t$.
To prove this, I know $A^t=A$ by symmetry, then $aA^t=aA$ and $bB^t=bB$. Since symmetric matrices are closed under condition, $aA+bB=aA^t+bB^t$ (do I need to elaborate more on this?). Apply transformation on both sides, get $(aA+bB)^t=aA^t+bB^t$.