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Problem Set 3 Problem 24

Problem Set 3 Problem 24. By Ellen Dickerson. Problem 24. If the matrix has 1 as an eigenvalue , find another eigenvalue for A. To find the eigenvalues of a 2x2 matrix we do the following 1.) Form the matrix A- λ I

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Problem Set 3 Problem 24

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  1. Problem Set 3 Problem 24 By Ellen Dickerson

  2. Problem 24 If the matrix has 1 as an eigenvalue, find another eigenvalue for A.

  3. To find the eigenvalues of a 2x2 matrix we do the following 1.) Form the matrix A- λI 2.) Solve the equation (a- λ)(d- λ) – bc = 0 for λ by factoring or using the quadratic formula - λ - (t- λ)(2t- λ) – (1)(1-t) =0

  4. (t- λ)(2t- λ) – (1)(1-t) = 0 We know one of the eigenvalues is 1 so we will replace lambda with 1. (t-1)(2t-1) – (1)(1-t) = 0 Now we will solve for t. 2t2 – 3t + 1 – (1 – t) = 0 2t2 – 2t = 0 t(2t - 2) = 0 t = 0 or 2t - 2 = 0 2t = 2 t = 1

  5. Lets say that t = 1 then is equal to So now we will find the eigenvalue for (1- λ)(2- λ) - (1)(0) = 0 λ 2 - 3 λ + 2 = 0 (λ-2)(λ-1) = 0 (λ-2) = 0 or (λ-1) = 0 λ = 2 or λ = 1 We already know 1 is one of the eigenvalues so 2 is another eigenvalue for the matrix.

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