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Inverses and determinants

Source: Romanian District Olympiad 2023 11.2

March 11, 2023
linear algebramatrixdeterminant

Problem Statement

Let AA{} and BB be invertible n×nn\times n matrices with real entries. Suppose that the inverse of A+B−1A+B^{-1} is A−1+BA^{-1}+B. Prove that det⁡(AB)=1\det(AB)=1. Does this property hold for 2×22\times 2 matrices with complex entries?
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