Thanks a lot!!!
In general, $\displaystyle det(A + B) \ne det(A) + det(B).$
Example: $\displaystyle A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\ B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix},\ det(A) = det(B) = 0,\ det(A+B) = 1. $
For square matrices, $\displaystyle det(AB) = det(A)det(B).$