Question

g Problem 2. (4 points) Suppose we convert the difference equation x(t+ 1) = 7x(t) + 8x(t−1), t = 1, 2, 3, · · · with x(0) = 1 and x(1) = 3 into the matrix form ? x(t + 1) x(t) ? = ? 7 8 1 0? ? x(t) x(t − 1)? , t = 1, 2, 3, · · · Which of the following statements must be TRUE? (I) The matrix A = ? 7 8 1 0? is diagonalizable, for example, with P = ? −1 8 1 1? , there exists a matrix D = ? −1 0 0 8? such that P −1AP = D. (II) Suppose we let u(t+1) = ? x(t + 1) x(t) ? , then there exists an 2 by 2 invertible matrix P and a diagonal matrix D such that u(t) = P Dt−1P −1 ? 3 1 ? (III) Suppose we let u(t + 1) = ? x(t + 1) x(t) ? , then u(t) = 5 9 (−1)t−1 ? −1 1 ? + 4 9 (8)t−1 ? 8 1 ? , t = 0, 1, 2, · · · (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III

Answer 1

From the equations given

7x(t)+8x(t-1)=x(t+1)

where t=1,2,3

and initial conditions are

x(0)=1

x(1)=3

From the given data

Option C is correct

II and III only