MathDB
sum_i sum_j a_ij=n where A is nxn

Source: VJIMC 2006 1.4

June 28, 2021
matrixlinear algebra

Problem Statement

Let A=[aij]n×nA=[a_{ij}]_{n\times n} be a matrix with nonnegative entries such that ∑i=1n∑j=1naij=n.\sum_{i=1}^n\sum_{j=1}^na_{ij}=n. (a) Prove that ∣det⁡A∣≤1|\det A|\le1. (b) If ∣det⁡A∣=1|\det A|=1 and λ∈C\lambda\in\mathbb C is an arbitrary eigenvalue of AA, show that ∣λ∣=1|\lambda|=1.
Back to ProblemsView on AoPS