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Finding Eigenvalues for a 2x2 Matrix with Known Eigenvalue

In this problem, we are tasked with determining another eigenvalue of a 2x2 matrix ( A ) given that one eigenvalue is 1. We begin by forming the characteristic equation for the matrix, which is derived from ( text{det}(A - lambda I) = 0 ). Using the known eigenvalue, we can substitute and solve the derived polynomial to find the remaining eigenvalues. Through careful calculations, we arrive at the conclusion that the second eigenvalue is 2, confirming that the eigenvalues of the matrix are 1 and 2.

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Finding Eigenvalues for a 2x2 Matrix with Known Eigenvalue

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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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