Answer
See below
Work Step by Step
1. Property:
$=a_{11}a_{22}+a_{12}a_{21}+a_{21}a_{12}+a_{22}a_{11}=2(a_{11}a_{22}+a_{12}a_{21})$
Assume $A=\begin{bmatrix}
0 & -1\\
1 & 1
\end{bmatrix}$ then $=2[0.1+(-1).1]=-2 \lt 0$
Hence, the property $=0 \leftrightarrow A=0$ does not hold.
2. Property
$=a_{11}b_{22}+a_{12}b_{21}+a_{21}b_{12}+a_{22}b_{11}\\
=b_{11}a_{22}+b_{12}a_{21}+b_{21}a_{12}+b_{22}a_{11}\\
=, \forall A,B \in M_2(R)$
3. Property
$=(ka_{11})b_{22}+(ka_{12})b_{21}+(ka_{21})b_{12}+(ka_{22})b_{11}\\
=k(a_{11}b_{22}+a_{12}b_{21}+a_{22}b_{11})\\
=k,\forall k \in R, \forall A,B \in M_2(R)$
4. Property
Assume $C=\begin{bmatrix}
c_{11} & c_{12}\\
c_{21} & c_{22}
\end{bmatrix}$
$=(a_{11}+b_{11})c_{22}+(a_{12}+b_{12})c_{21}+(a_{21}+b_{21})c_{12}+(a_{22}+b_{22})c_{11}\\
=(a_{11}c_{22}+a_{12}c_{21}+a_{21}c_{12}+a_{22}+c_{11})+(b_{11}+c_{22}+b_{12}c_{21}+b_{21}c_{12}+b_{22}c_{11})\\
=+, \forall A,B,C \in M_2(R)$