Solution: In Exercises 16, determine which of x1, x2, and x3 is an eigenvectorfor the

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. A = 1 3 2 2 , x1 = 3 2 , x2 = 1 1 , x3 = 2 2

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. A = 1 2 0 3 , x1 = 0 2 , x2 = 1 3 , x3 = 1 2

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. A = 2 72 0 1 0 0 2 1 , x1 = 3 1 1  , x2 = 2 0 1  , x3 = 1 0 0 

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. A =  3 1 0 1 30 1 12 , x1 = 1 1 1  , x2 = 1 1 0  , x3 =  1 2 1 

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. = 6 3 10 0 3 10 6 6 00 3 3 2 3 , x1 = 1 1 0 0 , x2 = 1 2 1 0 , x3 = 1 1 3 2

In Exercises 16, determine which of x1, x2, and x3 is an eigenvector for the matrix A. For those that are, determine the associated eigenvalue. A = 551 8 821 8 6 6 9 0 7 1 2 10 , x1 = 1 1 0 2 , x2 = 1 1 0 0 , x3 = 1 2 2 1

In Exercises 710, use the characteristic polynomial to determine if is an eigenvalue for the matrix A. A = 2 7 1 6 , = 3 8

In Exercises 710, use the characteristic polynomial to determine if is an eigenvalue for the matrix A. A = 2 7 1 6 , = 3 8

In Exercises 710, use the characteristic polynomial to determine if is an eigenvalue for the matrix A. A = 02 0 20 0 2 2 2  , = 2

In Exercises 710, use the characteristic polynomial to determine if is an eigenvalue for the matrix A. A =  6 3 1 4 1 1 18 6 4  , = 1

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 1 3 1 5 , = 4

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 2 4 3 1 , = 5 1

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 6 10 2 3 , = 2 1

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 11 12 8 9 , = 3 1

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 6 3 7 4 15 4 3 9 , = 4

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 2 5 7 2 11 13 2 5 7  , = 4

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 5 1 2 2 22 2 1 5 , = 6

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A =  1 2 7 10 2 2 10 2 2 , = 9

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 11 3 3 8 13 5 5 8 2 2 2 0 3 3 30 , = 4

In Exercises 1120, find a basis for the eigenspace of A associated with the given eigenvalue . A = 15 3 15 8 21 9 17 8 2 2 6 0 3 3 15 4 , = 8

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 2 0 4 3

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 2 6 1 1

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 1 2 2 3

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 2 8 1 4

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A =  30 0 12 0 4 5 1 

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 001 100 010

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 251 0 3 1 2 14 4

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A =  0 3 1 121 3 9 4 

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 1 000 5 200 0 310 2 011

In Exercises 2130, find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A. A = 0010 0100 1000 0001

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. A 2 2 matrix A with eigenvalues = 1 and = 2.

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. A 2 2 matrix A with eigenvalues = 3 and = 0.

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. . A 3 3 matrix A with eigenvalues = 1, = 2, and = 3.

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. . A 3 3 matrix A with eigenvalues = 1 (multiplicity 2) and = 4.

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. A 2 2 matrix that has no real eigenvalues.

FIND AN EXAMPLE For Exercises 3136, find an example that meets the given specifications. A 4 4 matrix that has no real eigenvalues.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. . An eigenvalue must be nonzero, but an eigenvector u can be equal to the zero vector.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. The dimension of an eigenspace is always less than or equal to the multiplicity of the associated eigenvalue.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If u is a nonzero eigenvector of A, then u and Au point in the same direction.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If 1 and 2 are eigenvalues of a matrix, then so is 1 + 2.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If A is a diagonal matrix, then the eigenvalues of A lie along the diagonal.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If 0 is an eigenvalue of an n n matrix A, then the columns of A span Rn.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If 0 is an eigenvalue of A, then nullity(A) > 0.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. Row operations do not change the eigenvalues of a matrix.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. If 0 is the only eigenvalue of A, then A must be the zero matrix.

TRUE OR FALSE For Exercises 3746, determine if the statement is true or false, and justify your answer. The product of the eigenvalues (counting multiplicities) of A is equal to the constant term of the characteristic polynomial of A.

Suppose that A is a square matrix with characteristic polynomial ( 3)3( 2)2( + 1). (a) What are the dimensions of A? (b) What are the eigenvalues of A? (c) Is A invertible? (d) What is the largest possible dimension for an eigenspace of A?

Suppose that A is a square matrix with characteristic polynomial ( 1)3( + 2)3. (a) What are the dimensions of A? (b) What are the eigenvalues of A? (c) Is A invertible? (d) What is the largest possible dimension for an eigenspace of A?

Let T : Rn Rn be given by T(x) = Ax. Prove that if 0 is not an eigenvalue of A, then T is onto.

Prove that if is an eigenvalue of A, then 4 is an eigenvalue of 4A.

Prove that if = 1 is an eigenvalue of an n n matrix A, then A In is singular.

Prove that if u is an eigenvector of A, then u is also an eigenvector of A2.

Prove that u cannot be an eigenvector associated with two distinct eigenvalues 1 and 2 of A.

Prove that if 5 is an eigenvalue of A, then 25 is an eigenvalue of A2.

If u = 0 was allowed to be an eigenvector, then which values of would be associated eigenvalues? (This is one reason why eigenvectors are defined to be nonzero.)

Suppose that A is a square matrix that is either upper or lower triangular. Show that the eigenvalues of A are the diagonal terms of A.

Let A be an invertible matrix. Prove that if is an eigenvalue of A with associated eigenvector u, then 1 is an eigenvalue of A1 with associated eigenvector u.

Let A = a b c d . Find a formula for the eigenvalues of A in terms of a, b,c, and d. (HINT: The quadratic formula can be handy here.)

Suppose that A and B are both n n matrices, and that u is an eigenvector for both A and B. Prove that u is an eigenvector for the product AB.

Suppose that A is an n n matrix with eigenvalue and associated eigenvector u. Show that for each positive integer k, the matrix Ak has eigenvalue k and associated eigenvector u.

Suppose that the entries of each row of a square matrix A add to zero. Prove that = 0 is an eigenvalue of A.

Suppose that A = a b c d , where a, b,c, and d satisfy a+b = c + d. Show that 1 = a + b and 2 = a c are both eigenvalues of A.

Suppose that A is a square matrix. Prove that if is an eigenvalue of A, then is also an eigenvalue of AT . (HINT: Recall that det(A) = det(AT ).)

Suppose that A is an n n matrix and c is a scalar. Prove that if is an eigenvalue of A with associated eigenvector u, then c is an eigenvalue of A c In with associated eigenvector u.

Suppose that the entries of each row of a square matrix A add to c for some scalar c. Prove that = c is an eigenvalue of A.

Let A be an n n matrix. (a) Prove that the characteristic polynomial of A has degree n. (b) What is the coefficient on n in the characteristic polynomial? (c) Show that the constant term in the characteristic polynomial is equal to det(A). (d) Suppose that A has eigenvalues 1, ... , n that are all real numbers. Prove that det(A) = 12 n.

C In Exercises 6770, find the eigenvalues and bases for the eigenspaces of A. A = 0 0 2 1 11 6 5 20 4 1 2 0 2 1

C In Exercises 6770, find the eigenvalues and bases for the eigenspaces of A. = 20 9 14 18 40 17 18 28 17 9 10 11 17 9 14 15

C In Exercises 6770, find the eigenvalues and bases for the eigenspaces of A. A = 10 0 1 3 3 23 1 6 3 2 24 0 1 9 9 14 0 1 5 5 10 0 1 3 3

C In Exercises 6770, find the eigenvalues and bases for the eigenspaces of A. A = 50 2 1 1 61 4 3 3 6 0 3 3 3 20 2 3 2 4 0 2 1 2