International Mathematics Competition for University Students

July 27 - Aug 2 2015, Blagoevgrad, Bulgaria

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Problem 1

1. For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$.

Does the same conclusion follow for matrices with complex entries?

Proposed by Zbigniew Skoczylas, Wroclaw University of Technology