Sum of two invertible matrices [duplicate]

If A and B are two n x n invertible matrices, would the matrix result from A+B be invertible?

I think it would because for a matrix to be invertible its determinant would have to be greater than 0, and if you add the determinants of two matrices greater than 0 you would have to get a non zero answer. But is there any way to prove this?

marked as duplicate by Dietrich Burde, Community♦Dec 19 '18 at 19:54

• 1+(-1)=0$\phantom{}$. – user1551 Dec 19 '18 at 19:49

The answer is generally no. For instance, consider $$A = \pmatrix{1&0\\0&1}, \quad B = \pmatrix{-1&0\\0&-1}$$

• I see, thank you for your explanation! – Finalsock23 Dec 19 '18 at 19:53