IMC 2015 Problem 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)