can I have $\det(A+B)=0$ if $\det(A)=0$ and $\det(B) \neq 0$?

Question

Given two square matrix $A$ and $B$ that $\det(A)=0$ and $\det(B) \neq 0$. Is it possibly that $\det(A+B)=0$ ?

I have tried numerically, it seems that this is impossible. However I don't know how to prove it because there is no relation between $\det(A+B)=0$ and $\det(A)$, $\det(B)$.

Answer 1

$$ \begin{bmatrix} 1&1\\ 1&1 \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix} + \begin{bmatrix} 0&1\\ 1&1 \end{bmatrix} $$

Answer 2

Try $A= \matrix {0&1\\0&0}, B=\matrix {1&0\\1&1}$

Answer 3

Yes. Consider

$$A=\begin{pmatrix}0&0\\0&-1\end{pmatrix}$$ $$B=I_2$$

Answer 4

$A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, $B = \begin{pmatrix} -1 & -1 \\ 0 & -1\end{pmatrix}$