Answer
False
Work Step by Step
Let $A=\begin{bmatrix}
1 & 0 & 0 \\0 & 1 & 0\\0 & 0 & 0
\end{bmatrix}$ and $B=\begin{bmatrix}
0 & 0 & 0 \\0 & 0 & 0\\0 & 0 & 1
\end{bmatrix}$ be $2 \times 2$ matrices.
The we have $det (A)=0\\
det (B)=0$
and $det(A+B)=det(I_3)=1$
Hence, the statement is false.