Question

Let A=\(\begin{pmatrix} 2 &-1  &-1 \\   1&0  &-1 \\   1&-1  &0  \end{pmatrix}\) and B=A–I. If ω=\(\frac{\sqrt3 i-1}{2}\), then the number of elements in the set {n∈{1,2,⋯,100}:\(A^n+(ωB)^n\)=A+B} is equal to _______.

Answer 1

Correct Answer: 17

Given matrices \(A=\begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}\) and \(B=A-I=\begin{pmatrix} 1 & -1 & -1 \\ 1 & -1 & -1 \\ 1 & -1 & -1 \end{pmatrix}\), \(ω=\frac{\sqrt3 i-1}{2}\). We need to find the number of integers \(n \in \{1,2,\ldots,100\}\) such that \(A^n+(ωB)^n=A+B\).

The condition \(A^n + (ωB)^n = A+B\) is satisfied if and only if matrix powers behave such that exponential characteristics reduce to the identity transformation at specific \(n\). Analyzing cyclic properties of eigenvalues can suggest insights about periodic behavior. Begin by examining eigenvalues of B.

Calculate eigenvalues of \(B\):

1. Characteristic polynomial \(\det(B-\lambda I)=0\).
2. For matrix \(B=\begin{pmatrix} 1 & -1 & -1 \\ 1 & -1 & -1 \\ 1 & -1 & -1 \end{pmatrix}\), characteristic polynomial is \(-(λ+1)^2(λ-3)=0\). Thus, eigenvalues are \(-1\) (multiplicity 2) and \(3\).

Thus, \(B\) is cyclic with a specific pattern, with minimal relation matrices connecting through periodic exponents.

Analyzing for \( (ωB)^n \), notice: \(
ω^3=1\), given \(|ω|=1\) and acts akin to complex rotation cycles within trigonometric properties, such that \(ω^n\) is periodic with period 3:

• \((ωB)^3=B^n\), for eigenvectors interchanging roles through powers as per complex conjugate multiplication cycles produces identity-like matrices.

Examine \((ωB)^{3k}\) terms:

1. \(\begin{cases}A^n,... \end{cases} \to (k \equiv 0)\).
2. Verify when \(A^n+(ωB)^n=(A+AB)\) holds \(n \equiv 0 \pmod{3}\).
3. Count for \(1\leq n \leq 100\), integers \(n\) divisible by 3: \(3,6,...,99\).

These constitute an arithmetic sequence. Sum constraint calculation: \((99-3)/3 +1 = 33\).

Thus, the number of such integers \(n\) is indeed 33, confirming expectations based on combinatorial characteristics, known eigenvalue periodicity through eigenstructures. The solution’s range verification: 33 satisfies bounded logical constraint, max | min dictates ≤ 100.