Sum of two invertible matrices [duplicate]

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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?

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1 Answer

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

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