MathDB

Problem 2

Part of 2007 VJIMC

Problems(2)

A+A^T=I implies det A>0

Source: VJIMC 2007 2.2

6/24/2021
Let AA be a real n×nn\times n matrix satisfying A+AT=I,A+A^{\text T}=I,where ATA^{\text T} denotes the transpose of AA and II the n×nn\times n identity matrix. Show that det⁡A>0\det A>0.
matrixlinear algebra
necklace, min. # of colorings to distinguish beads

Source: VJIMC 2007 1.2

6/24/2021
Alice has got a circular key ring with nn keys, n≥3n\ge3. When she takes it out of her pocket, she does not know whether it got rotated and/or flipped. The only way she can distinguish the keys is by coloring them (a color is assigned to each key). What is the minimum number of colors needed?
combinatorics