International Mathematics Competition
for University Students
2015

IMC2015: Day 1, 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

Hint: Multiply by $(A+B)$. Notice that $(AB^{-1})^{-1}=BA^{-1}$.

  Solution