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Let P be an orthogonal matrix and Q be a non-zero invertible matrix such that P^(-1) * Q * (P^T)^(-1) = [[1, 1], [0, 1]]. Find P^T * Q²⁰¹⁹ * P.

1. [[1, 0], [0, 1]]
2. [[0, 0], [0, 0]]
3. [[2019, 1], [0, 1]]
4. [[1, 2019], [0, 1]]

Correct answer: [[1, 2019], [0, 1]]

Solution

P is orthogonal, so P^(-1) = P^T, which means (P^T)^(-1) = P. The given equation: P^(-1) * Q * (P^T)^(-1) = M where M = [[1,1],[0,1]]. Substituting: P^T * Q * P = M. Therefore Q = P * M * P^T = P * M * P^(-1). Now compute P^T * Q²⁰¹⁹ * P: P^T * Q²⁰¹⁹ * P = P^T * (P*M*P^(-1))²⁰¹⁹ * P = P^T * P * M²⁰¹⁹ * P^(-1) * P = I * M²⁰¹⁹ * I = M²⁰¹⁹. Power of M: [[1,1],[0,1]]ⁿ = [[1,n],[0,1]]. Therefore M²⁰¹⁹ = [[1,2019],[0,1]].

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