Answer
(a) \begin{align*}
\operatorname{Tr}(A+B)&=\sum_{i=1}^{n} (a_{i i} +b_{ii})\\
&=\sum_{i=1}^{n} a_{i i} +\sum_{i=1}^{n}b_{ii}\\
&=\operatorname{Tr}(A)+\operatorname{Tr}(B).
\end{align*}
(b) \begin{align*}
\operatorname{Tr}(cA )&=\sum_{i=1}^{n} (ca_{i i} )\\
&=c\sum_{i=1}^{n} a_{i i} \\
&=c\operatorname{Tr}(A).
\end{align*}
Work Step by Step
(a) \begin{align*}
\operatorname{Tr}(A+B)&=\sum_{i=1}^{n} (a_{i i} +b_{ii})\\
&=\sum_{i=1}^{n} a_{i i} +\sum_{i=1}^{n}b_{ii}\\
&=\operatorname{Tr}(A)+\operatorname{Tr}(B).
\end{align*}
(b) \begin{align*}
\operatorname{Tr}(cA )&=\sum_{i=1}^{n} (ca_{i i} )\\
&=c\sum_{i=1}^{n} a_{i i} \\
&=c\operatorname{Tr}(A).
\end{align*}
