Answer
See below
Work Step by Step
Let $A=[a_{ij}]\\
B=[b_{ij}]\\
A+B=[a_{ij}+b_{ij}]\\
kA=[ka_{ij}$ $\in M_n(R)$
with $k \in R$ scalar.
$T:M_n(R) \rightarrow R$ defined by $T(A)=tr(A)=\sum^n_{i=1}a_{ii}$
We have:
$$T(A+B)=tr(A+B)\\
=\sum^n_{i=1}(a_{ii}+b_{ii})\\
=\sum ^n_{i=1}a_{ii}+\sum^n_{i=1}b_{ii}\\
=tr(A)+tr(B)\\
=T(A)+T(B)$$
$$T(kA)=tr(kA)\\
=\sum^n_{i=1}ka_{ii}\\
=k\sum^n_{i=1}a_{ii}\\
=ktr(A)\\
=kT(A)$$
Thus it is a linear transformation.