Suppose $A$, $B$ are real $n \times n$ matrices with $A + B = I$ and $\operatorname{rank} (A) + \operatorname{rank} (B) = n$.
How can one show that $AB = BA = 0$?
Two matrices of complementary rank that sum to the identity have zero product.
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linear-algebra
matrices
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0What have you tried? First, did you manage to get the easy part, that is, $AB=BA$? – 2012-10-28