6.3. EXERCISES 153

and its row echelon form is  1 0 0 0

0 1 1 0

0 0 0 0

 .

Therefore, the eigenvectors are of the form

v = z

 0

−1

1

 .

It follows  0

−1

1

 cos(√

3t),

 0

−1

1

 sin(√

3t)

are both solutions to the system of differential equations. You can find other oscillatorysolutions in the same way by considering the other eigenvalues. You might try checkingthese answers to verify they work.

This is just a special case of a procedure used in differential equations to obtain closedform solutions to systems of differential equations using linear algebra. The overall philos-ophy is to take one of the easiest problems in analysis and change it into the eigenvalueproblem which is the most difficult problem in algebra. However, when it works, it givesprecise solutions in terms of known functions.

6.3 Exercises

1. If A is the matrix of a linear transformation which rotates all vectors in R2 through30◦, explain why A cannot have any real eigenvalues.

2. If A is an n×n matrix and c is a nonzero constant, compare the eigenvalues of A andcA.

3. If A is an invertible n × n matrix, compare the eigenvalues of A and A−1. Moregenerally, for m an arbitrary integer, compare the eigenvalues of A and Am.

4. Let A,B be invertible n × n matrices which commute. That is, AB = BA. Supposex is an eigenvector of B. Show that then Ax must also be an eigenvector for B.

5. Suppose A is an n × n matrix and it satisfies Am = A for some m a positive integerlarger than 1. Show that if λ is an eigenvalue of A then |λ| equals either 0 or 1.

6. Show that if Ax = λx and Ay = λy, then whenever a, b are scalars,

A (ax+ by) = λ (ax+ by) .

Does this imply that ax+ by is an eigenvector? Explain.

7. Find the eigenvalues and eigenvectors of the matrix

 −1 −1 7

−1 0 4

−1 −1 5

 . Determine

whether the matrix is defective.