Let $A$ and $B$ be $2\times 2$ matrices such that $A^2 - B^2$ is invertible. Is $A-B$ necessarily invertible?
This doesn't seem like it should be difficult but I just can't come up with a solution.
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$\begingroup$ It is if $A$ and $B$ commute. $\endgroup$– JonathanZ supports MonicaCMay 21, 2020 at 16:16
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1 Answer
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The answer is no. For instance, consider $$ A = \pmatrix{0&1\\1&0}, \quad B = \pmatrix{0&1\\0&0}. $$
answered May 21, 2020 at 14:47
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1$\begingroup$ Thanks! I suspected the answer was no but couldn't find a counter example. $\endgroup$– kell99May 21, 2020 at 14:54