A matrix [tex]\mathbf X[/tex] is symmetric if [tex]\mathbf X=\mathbf X^\top[/tex]. We have
[tex](\mathbf B+\mathbf B^\top)^\top=\mathbf B^\top+(\mathbf B^\top)^\top=\mathbf B^\top+\mathbf B=\mathbf B+\mathbf B^\top[/tex]
and so [tex]\mathbf B+\mathbf B^\top[/tex] is indeed symmetric.
