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.
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.)
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.)