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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

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

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

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

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