If $A$ and $B$ are $2\times 2$ square matrices, then the determinant $\det (A+B)$ is generally:

Step 1 (Analyze additive determinant relation): In general matrix algebra, the determinant function is multiplicative (i.e., $\det (AB)=\det (A)\det (B)$) but is not additive:

Step 2 (Construct a counterexample): Let $A=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$ and $B=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$.

• $\det (A)=1\times 0-0\times 0=0$

• $\det (B)=0\times 1-0\times 0=0$

• Sum of determinants: $\det (A)+\det (B)=0+0=0$

Now compute $A+B$:

• $\det (A+B)=\det (I)=1$

Since $1\ne 0$:

Step 3 (Conclusion): Therefore, $\det (A+B)$ is generally not equal to $\det (A)+\det (B)$, corresponding to option index 0.