Page 576 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Chapter 7

Technology Exercises

The following exercises are designed to be solved using a technology utility. Typically, this will be MATLAB, Mathematica, Maple,
Derive, or Mathcad, but it may also be some other type of linear algebra software or a scientific calculator with some linear algebra
capabilities. For each exercise you will need to read the relevant documentation for the particular utility you are using. The goal of
these exercises is to provide you with a basic proficiency with your technology utility. Once you have mastered the techniques in
these exercises, you will be able to use your technology utility to solve many of the problems in the regular exercise sets.

Section 7.1

T1. (Characteristic Polynomial) Some technology utilities have a specific command for finding characteristic polynomials, and

in others you must use the determinant function to compute . Read your documentation to determine which

method you must use, and then use your utility to find for the matrix in Example 2.

T2. (Solving the Characteristic Equation) Depending on the particular characteristic polynomial, your technology utility may

or may not be successful in solving the characteristic equation for the eigenvalues. See if your utility can find the eigenvalues

in Example 2 by solving the characteristic equation .

T3.
(a) Read the statement of the Cayley–Hamilton Theorem in Supplementary Exercise 7 of this chapter, and then use your
technology utility to do that exercise.

(b) If you are working with a CAS, use it to prove the Cayley–Hamilton Theorem for matrices.

T4. (Eigenvalues) Some technology utilities have specific commands for finding the eigenvalues of a matrix directly (though the
procedure may not be successful in all cases). If your utility has this capability, read the documentation and then compute the
eigenvalues in Example 2 directly.

T5. (Eigenvectors) One way to use a technology utility to find eigenvectors corresponding to an eigenvalue is to solve the

linear system . Another way is to use a command for finding a basis for the nullspace of (if available).

However, some utilities have specific commands for finding eigenvectors. Read your documentation, and then explore

various procedures for finding the eigenvectors in Examples 5 and 6.

Section 7.2

T1. (Diagonalization) Some technology utilities have specific commands for diagonalizing a matrix. If your utility has this
capability, read the documentation and then use your utility to perform the computations in Example 2.
Note Your software may or may not produce the eigenvalues of A and the columns of P in the same order as the example.