Answer
See below
Work Step by Step
Let $A,B \in M_n(R)$
$\rightarrow (A+B)^T=A^T+B^T\\
\rightarrow (kA)^T=kA^T$
with $k \in R$ scalar.
$S:M_n(R) \rightarrow M_nR$ defined by
$S(A)=A+A^T$
We have:
$$S(A+B)=(A+B)+(A+B)^T\\
=A+B+A^T+B^T\\
=(A+A^T)+(B+B^T)\\
=S(A)+S(B)$$
$$S(kA)=(kA)+(kA)^T\\
=k(A+A^T)\\
=kS(A)\\
=ktr(A)$$
Thus it is a linear transformation.
