Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rrrr}{-2} & {1} & {0} & {0} \\ {1} & {0} & {0} & {1} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right)$$

Let A and B be

$$n \times n$$

matrices with eigenvalues

$$\lambda \text { and } \mu$$

respectively. (a) Give an example to show that

$$\lambda + \mu$$

need not be an eigenvalue of A + B. (b) Give an example to show that

$$\lambda \mu$$

need not be an eigenvalue of AB. (c) Suppose

$$\lambda \text { and } \mu$$

correspond to the same eigenvector

$$\mathbf x.$$

Show that, in this case,

$$\lambda + \mu$$

is an eigenvalue of A + B and

$$\lambda \mu$$

is an eigenvalue of AB.

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{llll}{5} & {1} & {0} & {9} \\ {0} & {1} & {0} & {9} \\ {0} & {0} & {0} & {9} \\ {0} & {0} & {0} & {0}\end{array}\right)$$

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rrr}{-3} & {1} & {1} \\ {0} & {0} & {0} \\ {0} & {1} & {0}\end{array}\right)$$

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rrr}{-14} & {1} & {0} \\ {0} & {2} & {0} \\ {1} & {0} & {2}\end{array}\right)$$

Choose the letter that best answers the question or completes the statement. The law of conservation of energy states that a. energy cannot be converted from one form to another. b. energy cannot be created or destroyed. c. energy resources must be used efficiently. d. energy is constantly being lost to friction.

a. Show that if $0<\theta<\pi$, then

$$A=\left[\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$

has no real eigenvalues and consequently no real eigenvectors.

b. Give a geometric explanation of the result in part (a).

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{cc}{1} & {-6} \\ {2} & {2}\end{array}\right)$$

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rr}{-5} & {0} \\ {1} & {2}\end{array}\right)$$

If 3 is an eigenvalue of an $n \times n$ matrix A, then 9 must be an eigenvalue of $A^{2}.$

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rr}{6} & {-2} \\ {-3} & {4}\end{array}\right)$$

Find the eigenvalues of the matrix. For the eigenvalue, find an eigenvalue. Sketch the Gershgorin circles, and then locate the eigenvalues in these circles.

$$\left( \begin{array}{rrr}{-2} & {1} & {0} \\ {1} & {3} & {0} \\ {0} & {0} & {-1}\end{array}\right)$$

Show that W, the set of all 3x3 upper triangular matrices, forms a subspace of all 3x3 matrices. What is the dimension of W? Find a basis for W.

Show that the value of the determinant

$$\left|\begin{array}{ccc}{\sin (\theta)} & {\cos (\theta)} & {0} \\ {-\cos (\theta)} & {\sin (\theta)} & {0} \\ {\sin (\theta)-\cos (\theta)} & {\sin (\theta)+\cos (\theta)} & {1}\end{array}\right|$$

is independent of $\theta$.