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If A= ( array ll 1 & 2 0 & 1 array ) P= ( array cc & - & array ) Q=P^T A P, then P Q²⁰¹⁴ P^T is equal to

Options

  1. A(P^T )²⁰¹³ A²⁰¹⁴ P²⁰¹³
  2. B( array cc 1 & 2²⁰¹⁴ 0 & 1 array )
  3. CP^T A²⁰¹⁴ P
  4. D( array cc 1 & 4028 0 & 1 array )

Correct answer

D. ( array cc 1 & 4028 0 & 1 array )

Step-by-step solution

Given Q = P^T A P . We need to evaluate P Q²⁰¹⁴ P^T . Note that Q^2 = (P^T A P)(P^T A P) = P^T A (P P^T) A P . Since P is an orthogonal matrix, P P^T = I , where I is the identity matrix. Thus, Q^2 = P^T A^2 P . By induction, Q^n = P^T A^n P for any positive integer n . Therefore, Q²⁰¹⁴ = P^T A²⁰¹⁴ P . Substituting this into the expression P Q²⁰¹⁴ P^T , we get P (P^T A²⁰¹⁴ P) P^T . Using the associative property of matrix multiplication, (P P^T) A²⁰¹⁴ (P P^T) = I A²⁰¹⁴ I = A²⁰¹⁴ . For the matrix A = pmatrix 1 & 2 0

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