IMC 2015 · Problem 1

Day 122nd IMC · Blagoevgrad, Bulgaria

Statement

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?

Hidden so you can work on the problem first.

Proposed by Zbigniew Skoczylas, Wrocªaw University of Technology.