If $A=B+C$, where $B=\begin{bmatrix}a &b &c\\ d& e& f\\ g &h& i\end{bmatrix}$ and $C=\begin{bmatrix}k &k &k\\ 0& 0& 0\\ 0 &0& 0\end{bmatrix}$, then $|A|=|B|+|C|$. Since $|C|=0$, so, $|A|=|B|$.
But, if we consider the properties of determinants, then $\left| \begin{array}{c c c} a+k & b+k & c+k \\ d & e & f \\ g & h & i \end{array}\right|=\left| \begin{array}{c c c} a & b & c \\ d & e & f \\ g & h & i \end{array}\right|+\left| \begin{array}{c c c} k & k & k \\ d & e & f \\ g & h & i \end{array}\right|$. Thus, $|A|\ne|B|$.
What is wrong in the first method?
It is false that $|B+C| = |B| + |C|$.
The true result, is that the determinant of a product is the product of the determinants. In formulae, $|BC| = |B|\cdot |C|$.