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Part of 2015 IMC

Problems(1)

IMC2015, problem 1

Source: IMC2015

7/29/2015
For any integer n≥2n\ge 2 and two n×nn\times n matrices with real entries A,  BA,\; B that satisfy the equation A−1+B−1=(A+B)−1  A^{-1}+B^{-1}=(A+B)^{-1}\; prove that det⁡(A)=det⁡(B)\det (A)=\det(B).
Does the same conclusion follow for matrices with complex entries?
(Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)
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