According to the result stated in Problem 1 and 2, the characteristic polynomial of 3 x 3
Question:
According to the result stated in Problem 1 and 2, the characteristic polynomial
of 3 x 3 matrix A is given by
The remaining coefficient c1 can be found by substituting ? = 1 and then calculating the two determinants |A| and p(1) = |A - 1|. Use this method to find the characteristic equation, eigenvalues, and associated eigenvectors of the matrix.
Problem 1:
Suppose that the characteristic equation IA - AAI = 0 is written as a polynomial equation (Eq. (5)). Show that the constant term is co = det A. Suggestion: Substitute an appropriate value for A.
Problem 2:
If A = [aij] is an n x n matrix, then the trace Tr A of A is defined to be
The sum of the diagonal elements of A. It can be proved that the coefficient of ?n - 1 in Eq. (5) is cn -1 = (-1) n - 1 (Tr A). Show explicitly that this is true in the case of a 2 x 2 matrix.
College Algebra Graphs and Models
ISBN: 978-0321845405
5th edition
Authors: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna