MathDB

2

Part of 2022 VJIMC

Problems(2)

A^7+A^5+A^3+A=I implies det(A)>0

Source: VJIMC 2022 1.2

4/11/2022
Let n≥1n\ge1. Assume that AA is a real n×nn\times n matrix which satisfies the equality A7+A5+A3+A−I=0.A^7+A^5+A^3+A-I=0. Show that det⁡(A)>0\det(A)>0.
matrixlinear algebra
irreducibility of a polynomial

Source: VJIMC 2022 2.2

4/11/2022
For any given pair of positive integers m>nm>n find all a∈Ra\in\mathbb R for which the polynomial xm−axn+1x^m-ax^n+1 can be expressed as a quotient of two nonzero polynomials with real nonnegative coefficients.
algebrapolynomial