True or false? For all $A,B \in \mathbb{R}^{n \times n}$ we have that det(A+B) $\neq$ det(A)$+$ det(B)

Question

Let $n \in \mathbb{N}$

True or false? For all $A,B \in \mathbb{R}^{n \times n}$ we have that $\det(A+B) \neq \det(A)+ \det(B)$.

The statement should be false because whenever we get that $\det(A) =0$ and $\det(B) =0$, we will have that $\det(A+B) = \det(A) + \det(B)$.

More specific counter-example, let $n = 0$.

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Is it alright like that or is my reasoning wrong?

Answer 1

Your reasoning is right, but you can not fix $n=0$ if you want to prove the statement is false for all $n$.

You are right when you say a counter-example is sufficient.

For example, take the null matrix $A$ of $\mathbb R^{n\times n}$, and $B:=A$.

You have $\det(A+B)=0=\det (A)+\det(B)$, so the statement is false.