MathDB

Problem 2

Part of 2012 VJIMC

Problems(2)

A in M2(prime), A=B^2 and det(B)=p^2

Source: VJIMC 2012 1.2

5/31/2021
Determine all 2×22\times2 integer matrices AA having the following properties:
1.1. the entries of AA are (positive) prime numbers, 2.2. there exists a 2×22\times2 integer matrix BB such that A=B2A=B^2 and the determinant of BB is the square of a prime number.
matrixlinear algebra
eigenvalues of tridiagonal matrix

Source: VJIMC 2012 2.2

5/31/2021
Let MM be the (tridiagonal) 10×1010\times10 matrix M=(−130⋯⋯⋯032−10⋮0−12−1⋱⋮⋮0−12⋱0⋮⋮⋱⋱⋱−10⋮0−12−10⋯⋯⋯0−12)M=\begin{pmatrix}-1&3&0&\cdots&\cdots&\cdots&0\\3&2&-1&0&&&\vdots\\0&-1&2&-1&\ddots&&\vdots\\\vdots&0&-1&2&\ddots&0&\vdots\\\vdots&&\ddots&\ddots&\ddots&-1&0\\\vdots&&&0&-1&2&-1\\0&\cdots&\cdots&\cdots&0&-1&2\end{pmatrix}Show that MM has exactly nine positive real eigenvalues (counted with multiplicities).
matrixlinear algebra