Answer
See below
Work Step by Step
1. Property:
$=\det (AA)=\det A.\det A=(\det A)^2 \geq 0$
Assume $A=\begin{bmatrix}
1 & 1\\
0 & 0
\end{bmatrix}$ then $=\det A.\det A=0.0=0$
but $A$ should not be equal to $0$.
Hence, the property $=0 \leftrightarrow A=0$ does not hold.
2. Property
$=\det (AB)=\det A.\det B=\det B.\det A=\det (BA)=, \forall A,B \in M_2(R)$
3. Property
$=\det((kA)B)=\det (kA)\det B=k^2\det A.\det B=k^2\det (AB)=k^2\det (AB)=k^2\ne k,\forall k \in R, \forall A,B \in M_2(R)$
Hence, the property does not hold.
4. Property
Assume $=\det ((A+B)+C)=\det (AC+BC)\ne+, \forall A,B,C \in M_2(R)$
Hence, the property does not hold.
