Does this statement hold true about square matrices $A$ and $B$:

Question

Does this statement hold true about square matrices $A$ and $B$:

If $\det (A) \neq 0$ and $\det(B) \neq 0$, then $\det(A+B) \neq 0$ or $\det(A-B) \neq 0$.

I tried researching about this but it seems not to be asked online.

Answer 1

It is not the case. Counterexample:

If

$A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \tag 1$

and

$B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \tag 2$

then

$\det(A) = -1 = -\det(B), \tag 3$

but

$A + B = \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}, \tag 4$

and

$A - B = \begin{bmatrix} 0 & 0 \\ 0 & -2 \end{bmatrix}, \tag 5$

so that

$\det (A + B) = \det (A - B) = 0. \tag 6$