# Linear Algebra : Eigenvalues and Eigenvectors

## Example Questions

### Example Question #451 : Operations And Properties

is an involutory matrix.

True, false, or indeterminate: 0 is an eigenvalue of .

True

False

Indeterminate

False

Explanation:

An eigenvalue of an involutory matrix must be either 1 or . This can be seen as follows:

Let  be an eigenvalue of involutory matrix . Then for some eigenvector ,

Premultiply both sides by :

By definition, an involutory matrix has  as its square, so

By transitivity,

Thus, , or

It follows that . The statement is false.

### Example Question #71 : Eigenvalues And Eigenvectors

The trace of a singular  matrix is 12. Give its set of eigenvalues.

Insufficient information is given to answer the question.

Explanation:

, being a singular matrix, must have 0 as an eigenvalue. Let  be its other eigenvalue..

The trace of a matrix is equal to the sum of its eigenvalues, so

,

and

The set of eigenvalues of  is .

### Example Question #72 : Eigenvalues And Eigenvectors

matrix  has  as its set of eigenvalues.

True, false, or indeterminate: the matrix is singular.

False

Indeterminate

True

False

Explanation:

A matrix is singular - that is, not having an inverse - if and only if one of its eigenvalues is 0. Since 0 is not an element of its eigenvalue set,  is nonsingular.

### Example Question #73 : Eigenvalues And Eigenvectors

matrix has as its set of eigenvalues .

True, false, or indeterminate: the matrix is singular.

Indeterminant

True

False

True

Explanation:

A matrix is singular - that is, not having an inverse - if and only if one of its eigenvalues is 0. This is seen to be the case.

### Example Question #74 : Eigenvalues And Eigenvectors

The trace of a singular  matrix  is 0.

Which of the following must be true of the eigenvalues of  as a result?

One eigenvalue is 0; the other two are each other's additive inverse.

0 is not an eigenvalue.

One eigenvalue is 0; the other two are each other's multiplicative inverse.

One eigenvalue is 0; the other two are each other's complex conjugate.

The only eigenvalue is 0.

One eigenvalue is 0; the other two are each other's additive inverse.

Explanation:

is singular, so the matrix must have 0 as an eigenvalue.

Let  be the other two eigenvalues. The sum of the eigenvalues of a matrix is equal to its trace, so

and

or

It follows that one eigenvalue must be 0, and the other two must be additive inverses.

### Example Question #76 : Eigenvalues And Eigenvectors

The trace of a singular  matrix  is 0; one of its eigenvalues is . What is it characteristic equation?

Explanation:

, being a singular matrix, must have 0 as an eigenvalue; it also has  as an eigenvalue. Being , it will have one more; call this eigenvalue

The sum of the eigenvalues of a matrix is equal to its trace, so

The set of eigenvalues is . The eigenvalues of a matrix are the solutions of its characteristic (polynomial) equation, which, as a consequence, is

### Example Question #75 : Eigenvalues And Eigenvectors

True or false: 0 is an eigenvalue of .

True

False

False

Explanation:

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of by adding the alternating products of each entry in any row or column and the corresponding adjoint. The third column is the easiest to do this with:

Since , 0 is not an eigenvalue of .

### Example Question #76 : Eigenvalues And Eigenvectors

True or false: 0 is an eigenvalue of .

False

True

True

Explanation:

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of by adding the alternating products of each entry in any row or column and the corresponding adjoint. The first row is the easiest to do this with:

Since  has zero as an eigenvalue.

### Example Question #79 : Eigenvalues And Eigenvectors

Calculate so that has 0 as an eigenvalue.

Explanation:

A necessary and sufficient condition for a matrix to have 0 as an eigenvalue is for the matrix to have determinant 0. Find the determinant of in terms of  by taking the product of the main diagonal elements and subtracting the product of the other two:

Set this equal to 0 and solve for :

.

### Example Question #80 : Eigenvalues And Eigenvectors

Calculate so that has 2 as an eigenvalue.