Answer
See below
Work Step by Step
Let $A,B \in M_2(R)$
$T:M_2(R) \rightarrow M_2R$ defined by
$T(A)=A^2$
We obtain:
$$T(A+B)=(A+B)^2\\
=(A+B)(A+B)\\
=AA+AB+BA+BB\\
=A^2+AB+BA+B^2\\
\ne A^2+B^2 =T(A)+T(B)$$
$$T(A+B) \ne T(A)+T(B)$$
Thus it is not a linear transformation.