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Eigenvalues and Eigenvectors

Part 3: Eigenvalues and Determinants

  1. For each matrix A, B, and C, compare the list of eigenvalues with its corresponding determinant. Formulate a rule relating the eigenvalues of a matrix to its determinant.
  2. Check this rule by entering a random 3 x 3 symmetric matrix and calculating both the eigenvalues and the determinant. (A matrix is symmetric if it is equal to its transpose. All the eigenvalues of a symmetric matrix are real.)
  3. Give an algebraic justification for your rule. (Hint: Look at the constant term in the characteristic polynomial -- both from the definition in terms of the determinant and as the product of linear factors of the characteristic polynomial.)

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