Answer
See below
Work Step by Step
1. Property:
$=a_{11}.a_{11}=a_{11}^2\geq0, \forall A \in M_2(R)\\
=0 \rightarrow a_{11}^2=0 \rightarrow a_{11}=0$
Assume $A=\begin{bmatrix}
0 & a_{12}\\
a_{13} & a_{14}
\end{bmatrix}=0$ but $A$ should not be equal to $0$.
Hence, the property $=0 \leftrightarrow A=0$ does not hold.
2. Property
$=a_{11}b_{11}=b_{11}a_{11}=, \forall A,B \in M_2(R)$
3. Property
$=(ka_{11})b_{11}=k(a_{11}b_{11})=k,\forall k \in R, \forall A,B \in M_2(R)$
4. Property
Assume $C=\begin{bmatrix}
c_{11}& c_{12}\\
c_{13} & c_{14}
\end{bmatrix} \in M_2(R)\\
\rightarrow =(a_{11}+b_{11})c_{11}=a_{11}b_{11}+b_{11}c_{11}=+, \forall A,B,C \in M_2(R)$
Consequently, 2.,3. and 4. properties do hold.