Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 &...
Question:
Determine if the following is true about {eq}A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix} {/eq}.
{eq}\lambda = -1 {/eq} is an eigenvalue. Its eigenvector satisfies {eq}-2x_1 = x_3 {/eq}
Eigenvector Satisfying a Matrix:
Given an eigenvalue of a square matrix, the eigenvector that satisfy the matrix and the eigenvector may be obtained from the augmented matrix of the null space involving the matrix and the eigenvalue.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerSolve the system
$$(A-\lambda\,I)x=0\\ $$
For {eq}\lambda=-1{/eq}, the augmented matrix is given by:
$$\left[\begin{array}{ccc|c} -1-(-1) & 4 &...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
Eigenvectors & Eigenvalues | Overview, Equations, & Examples
from
Chapter 6 / Lesson 3
2.8K
Understand the concept of eigenvalues of matrices and their corresponding eigenvectors. Learn the methods for finding eigenvalues and eigenvectors with examples.
Related to this Question
- Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix}. \lambda = 3 is an eigenvalue. Its eigenvector satisfies x_1 = -2x_2
- Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix}. \lambda = 3 is an eigenvalue. Its eigenvector satisfies x_2 = 3x_3
- Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix}. \lambda = 1 is an eigenvalue. Its eigenvector satisfies 2x_1 = 3x_2
- Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix}. \lambda = -1 is an eigenvalue. Its eigenvector satisfies x_1 = x_3
- Determine if the following is true about A = \begin{pmatrix} -1 & 4 & 0\\ 2 & 1 & -2\\ -4 & 4 & 3 \end{pmatrix}. \lambda = 1 is an eigenvalue. Its eigenvector satisfies -2x_1 = 3x_3
- Let A = (2, - 5; k, 7). Find the condition on k so that Lambda = - 1 is an eigenvalue of A. (Hint; Use the fact that Lambda is an eigenvalue if and only is det(A Lambda I) = 0.)
- Show that v is an eigenvector of A and find the corresponding eigenvalue, lambda. A = (1 2, 2 1), v = (4, -4).
- Determine if the given lambda is an eigenvalue of the matrix. [-7 -6 4 1 7 0 10 6 1], lambda = 3
- Let X be an eigenvector of A with eigenvalue λ. Show that X is an eigenvector of A^3 - 7A^2 + I and find its eigenvalue.
- Determine the eigenvalue for which \begin{bmatrix}-2 -2i\\-2\end{bmatrix} is an eigenvector of \begin{bmatrix}-6&4\\-2&-2\end{bmatrix}. a) 2 + 2i b) -2 + 2i c) -4 - 2i d) 10 + 2i
- Prove that if \lambda is an eigenvalue of a matrix A with corresponding eigenvector x, then \lambda^2 is an eigenvalue of A^2 with corresponding eigenvector x.
- Show that \lambda is an eigenvalue of A and find one eigenvector \vec v corresponding to this eigenvalue. A = \begin{bmatrix} 5 & 1 & -1 \\ 1 & 3 & 1 \\ 4 & 2 & 2 \end{bmatrix}, \ \lambda = 4.
- Let A (3 \times 3 matrice) = \begin{bmatrix} 0&1&0\\2&0&1\\0&-1&0 \end{bmatrix} . (i) Verify that 1 is an eigenvalue of A. (ii) Find an eigenvector of A corresponding to the eigenvalue 1.
- Determine the eigenvector of the matrix (1, 0, 1; 0, - 2, 0; - 2, 0, - 2) when the eigenvalue is 2.
- Determine the eigenvector of the matrix (1, 0, 1; 0, - 2, 0; - 2, 0, - 2) when the eigenvalue is 1.
- Determine the eigenvector of the matrix (1, 0, 1; 0, - 2, 0; - 2, 0, - 2) when the eigenvalue is 0.
- If lambda be an eigenvalue of a square matrix A, then prove the following.
- Suppose A has eigenvalue lambda = 2 with corresponding eigenvector u = (4 -3), and eigenvalue lambda = -1 with corresponding eigenvector v = (-2 3). Find the following. a. A^10 u. b. A^10 v.
- If \lambda is an eigenvalue of A , prove that \frac{1}{\lambda} is an eigenvalue of A^{-1}.
- If lambda =4 is an eigenvalue of the matrix (6,-1,2,3), then find the corresponding eigenvector.
- Let A = \begin{bmatrix} 1 & -1 & 0\\ -1 & 2 & -1\\ 0& -1&1 \end{bmatrix}. If A has 3 real eigenvalues, find the eigenvalue for the eigenvector (1, -2, 1).
- Is lambda = 4 an eigenvalue of [3 0 -2 2 3 3 -3 4 2]? If so. find one corresponding eigenvector.
- If lambda =3 is an eigenvalue of the matrix (2,-1,3,6), then find the corresponding eigenvector.
- Suppose A is a 2 \times 2 real matrix with an eigenvalue \lambda = 5 + 3i and corresponding eigenvector \vec{v} = \begin{bmatrix} -1+i\\ i \end{bmatrix}.
- Show that if lambda is an eigenvalue of an invertible matrix A, then 1 / lambda is an eigenvalue for A^{-1}.
- Find the eigenvalues and the corresponding eigenvectors for the given matrix \begin{pmatrix} 1 & -2 &0 \\ -2 & -1 &1 \\ 0& 0 & -1 \end{pmatrix}
- If \lambda is an eigenvalue of A, is it also an eigenvalue of A^T? What about the eigenvectors? Justify your answers.
- Find the basic eigenvectors of A corresponding to the eigenvalue lambda A = [-17 0 -15 -2 2 -2 20 0 18], lambda = 3
- Let A = \begin{bmatrix} 18 & -9 & a_{13} & 21\\ 12 & 3 & a_{23} & 15\\ -3 & -6 & a_{33} & 9\\ 12 & -9 & a_{43} & 27 \end{bmatrix}. Find the eigenvalue of A corresponding to the eigenvector \begin{bmatrix} 2\\ 2\\ 0\\ 2 \end{bmatrix}.
- Let A = \begin{bmatrix} 1 & 1\\ 1 & 1 \end{bmatrix} . (a) Find the eigenvalues of A. (b) For each eigenvalue, find an eigenvector.
- Find the eigenvalue and eigenvector of the given matrix.
- Consider the matrix. The eigenvalues of A are lambda 1 = 6 and lambda 2 = 4. Find the eigenvector associated with each eigen value.
- solve the eigenvalue problem y'' + lambda y = 0, y(0) = 0, integral_{0}^{L} y(x) d x = 0.
- \lambda is an eigenvalue of a matrix A if A - \lambdaI has linearly independent columns. True False
- Let A = (- 2 - 10 2 7). This matrix has real eigenvalues lambda _1 and lambda _2, where lambda _1 is smaller than lambda _2. (a) Find lambda _1 and lambda _2 and corresponding eigenvalues v_1 and
- The matrix A = [-2 -2 2 2], has an eigenvalue lambda of multiplicity 2 with corresponding eigenvector v. Find lambda and v.
- Solve for the eigenvector v associated by the eigenvalue 2 for the given matrix A.
- Let A = \begin{bmatrix} 6 &1 \\ 2 &7 \end{bmatrix} . (a) Find the eigenvalues of A. (b) Find a corresponding eigenvector for each eigenvalue in part(a).
- Here let A = \begin{bmatrix} 4 &14& 4\\ 1 & -1 &2\\-4 &-8 & -6\end{bmatrix}; v = \begin{bmatrix} 3 \\ -1 \\ -1 \end{bmatrix} (a) Show that v is an eigenvector for A and find the corresponding eigenvalue. (b) In fact, λ = 2 is another eigenval
- Calculate the Eigenvalue and Eigenvector of the following matrix. A = \begin{bmatrix} -5& -6 & -6\\ -1& 4 & 2\\ 3& -6 & -4 \end{bmatrix}
- Show that v is an eigenvector of A and find the corresponding eigenvalue. A = \begin{bmatrix}4 &- 2\\ 5& - 7\end{bmatrix}; V = \begin{bmatrix}1\\1\end{bmatrix}.
- For the following eigenvector \begin{bmatrix} -1\\ 0\\ -1 \end{bmatrix}, find the corresponding eigenvalue given the matrix\begin{bmatrix} -1 &3 &0 \\ 0& -4 &0 \\ -5& 3 &4 \end{bmatrix}.
- Find all of the eigenvalues \lambda of the matrix A. A=\begin{bmatrix} 2& 5 \\ 7 & 0 \end{bmatrix}
- Find the eigenvalue and eigenvector for the matrix. 2& 9 & 3 -2& -5 &0 2&6 &1
- Let T = \begin{pmatrix} 3 & -1 & 2 \\ 0 & 1 & -2\\ 0 & 0 & -1 \end{pmatrix} A) Find the eigenvalues and corresponding eigenvectors for T. B) Find the matrix C, such that C^1 TC=D, where D is
- Solve the eigenvalue problem {d^2}/{dx^2} (f(x)) = lambda f(x), f'(0) = 0 = f'(L) .
- Let x = 1 ? 1 2 , and let A = 3 ? 1 1 ? 1 3 ? 1 2 ? 2 4 . (a) Show that 2 is an eigenvalue of A , and find an associated eigenvector. (b) Show that x is an eigenvector of A , and find the a
- Give the eigenvalues and corresponding eigenvectors of A, A^{T}, A^{-1} for A = \begin{pmatrix} -1 & 0 & 1\\ 0 & 1 & 0\\ -3 & 2 & 3 \end{pmatrix}.
- The eigenvalues of the following system are \lambda = -2, -1, 3. Find the eigenvector for \lambda = 3. X' = \bigl(\begin{smallmatrix} -1 &1 &0 \\ 1& 2 &1 \\ 0& 3 &-1 \end{smallmatrix}\bigr)X
- How to find the eigenvector given eigenvalue?
- Find all real eigenvalues, with their algebraic multiplicities of the following matrix A = \begin{pmatrix} 3 & -2 & 5\\ 1 & 0 & 7\\ 0 & 0 & 2 \end{pmatrix}.
- Find the eigenvalues and eigenvectors of the following matrices:
- Let A be in R^(n x n) (1) Prove that if lambda in C not equal to 0 is an eigenvalue of A and A is invertible, then 1/lambda in C is an eigenvalue of A^(-1). (2) Prove that if lambda in C is an eig
- Let A be an invertible n by n matrix. Let be an eigenvalue of A, with corresponding eigenvector v . Prove that 1 is an eigenvalue of A 1 with corresponding eigenvector v. .
- Find all eigenvalue and eigenvectors of the given matrix. Show the eigenvectors as column vectors. \begin{bmatrix} -\frac{4}{17} & -\frac{2}{17} & -\frac{2}{17}\\ \frac{3}{17} & -\frac{3}{17} & -\frac{2}{17}\\ -\frac{3}{17} & \frac{5}{17} & \frac{4}{17}
- A= [ 1 1,1 1]. Solve the eigenvalue problem for the following matrix. Find all eigenvalues and eigenvectors
- 1. For each matrix and vector below, determine if the vector is an eigenvector for the given matrix. If it is, determine what the corresponding eigenvalue is. (a) \begin{bmatrix} 2 &-4 \\ 3&-6 \end{bmatrix}and, v=\begin{bmatrix} 2 & \\ 1& \end{bmat
- Find all the eigenvalues and corresponding eigenvectors of the following matrix
- Find the eigenvalues and eigen vector corresponding to each eigenvalue for the matrix A = \begin{bmatrix} -2 & 2\\ 1 & -3 \end{bmatrix}
- Find all 2 x 2 matrices for which the vector \begin{pmatrix}-1 \\ -2\end{pmatrix} in an eigenvector with associated eigenvalue -5.
- For the following eigenvector \begin{bmatrix} -1\\ 1\\ -1 \end{bmatrix}, find the corresponding eigenvalue given the matrix \begin{bmatrix} -1 &3 &0 \\ 0& -4 &0 \\ -5& 3 &4 \end{bmatrix}.
- Find all nonnegative eigenvalues and their associated eigenvectors for: y'' + lambda y = 0; y (0) = y' (pi) = 0.
- How to find the eigenvalue from the eigenvector?
- λ1 = 3 + 2i is an eigenvalue of A = \begin{pmatrix} 4&-5\\1&2\end{pmatrix} (a) Give the other eigenvalue λ2 of A. (b) Find the eigenvectors corresponding to λ1 and λ2. (c) Give the general solution of x' = Ax.
- A) Compute the eigenvalues and eigenvectors of the following matrices: (i) A = (4 -2, 5 -3) (ii) A = (6 1, -3 2) (iii) A = (5 3, -3 -1) B) Given that one eigenvalue is lambda = -4, compute the remaini
- For each matrix and vector below, determine if the vector is an eigenvector for given matrix. If it is, determine what the corresponding eigenvalue is. (a) A = \begin{bmatrix} 2 &-4 \\ 3 & -6 \end{bmatrix} and v = \begin{bmatrix} 2 \\ 1 \end{bmatr
- Let \begin{pmatrix} -2 &10\\2& 7 \end{pmatrix}. This matrix has real eigenvalues l\ambda_{1} and \lambda_{2}, where \lambda_{1} <\ lambda_{2}.a)Find \lambda_{1} and \lambda_{2} and corresponding eigen
- Multiply the matrix and the vector to determine if the vector is an eigenvector. if so,what is the eigenvalue? \begin{bmatrix} 5 & 0\\ 3 & -2 \end{bmatrix} \times \begin{bmatrix} 1 \\ 0 \end{bmatri
- An eigenvector belonging to the eigenvalue 1 + 3i of the matrix \begin{pmatrix} -2 & -18\\ 1 & 4 \end{pmatrix} is: (a)\; \begin{pmatrix} 6i\\ 1 - i \end{pmatrix}\\ (b)\; \begin{pmatrix} -6i\\ 1 + i \end{pmatrix}\\ (c)\; \begin{pmatrix} 1\\ -3 + 3i \en
- Find the eigenvalues and eigenvectors of the matrix. A = \begin{bmatrix} 1 &2\\ 2& 4 \end{bmatrix}
- Find the basic eigenvectors of A = (2 0 0 0 17 9 0 -30 -16) corresponding to the eigenvalue lambda= -1
- Find the basic eigenvectors of A= (1 -6 3 0 7 -3 0 14 -6) corresponding to the eigenvalue lambda=1.
- Find the eigenvalues and eigenvectors of the given matrices A and B.
- Let B = \begin{bmatrix} 4&0 \\ 0&5 \end{bmatrix} What are the eigenvectors and eigenvalues for B? Let C = \begin{bmatrix} \cos(\pi/4)& - \sin(\pi/4) \\ \sin(\pi/4)& \cos(\pi/4) \end{bmat
- Given the matrix [ A ] = [ 10 4 4 12 ] and the eigenvalues ?1 = 11 + r 2 = 11 ? r and the eigenvectors { v 1 } = { 4 1 + r } { v 2 } = { 4 1 ? r } , verify that [ A ] { v 1 } = ? 1 { v 1 } [ A ]
- a) Show that if \lambda is an eigenvalue for a matrix M, then \lambda^2 is an eigenvalue for the matrix M^2
- Let ??^ X ? = ?^ A ?^ X , , where ?^ A is a ( 2 � 2 ) matrix. suppose one eigenvalues of ?^ A is ? = 1 + 2 i and that the corresponding eigenvector is ?^ u = ( 1 i ) what is the homogenous
- Suppose A is an invertible n x n matrix and v is an eigenvector of A with associated eigenvalue -3.
- Find the eigenvalues and eigenvector of the matrix. A = ((1 2 -1), (1 0 1), (4 -4 5))
- Find the eigenvalues and an eigen vector corresponding to each eigenvalue for the matrix A = \begin{bmatrix} 1 & -4\\ 4 & -7 \end{bmatrix}
- Find the eigenvalues and eigenvectors of the matrix \begin{bmatrix} 6 & 4 &4 \\ -7 & -2&-1 \\ 7 &4 & 3 \end{bmatrix}
- Find the eigenvalues and corresponding eigenvectors of the given matrix. Then, determine whether the matrix is diagonalizable. \displaystyle{ A = \left[ \begin{array}{rrr} 2 & 1 & 1 \\ 2 & 3 & 2 \\ 1 & 1 & 0 \end{array} \right] }
- Find eigenvalues and eigenvectors of the following matrix A, and then find A^n. A = [3 -1 0, -1 2 -1, 0 -1 3].
- Consider the given matrix. \begin{pmatrix} 3 & -1 & 0 \\ 0 & -3 & 5\\ 3 & -1 & 0 \end{pmatrix} Find the eigenvalues. Find the eigenvectors.
- Consider the given matrix. A. Find the eigenvalues. B. Find the eigenvectors.
- Suppose the eigenvalues of A are 1,1,2. True or False? a) A is invertible b) A is diagonalizable c) A is not diagonalizable
- How to find eigenvalues when eigenvector and the corresponding matrix is given?
- Find the eigenvalues and corresponding eigenvectors of the matrix A = \begin{bmatrix} 20 & 30\\ -12 & -16 \end{bmatrix}
- Find all eigenvalues and eigenvectors of the following matrices. A= \begin{bmatrix} 0&-3 &3 \\ 1&-3 &0 \\ 1&0 &-3 \end{bmatrix}\ B=\begin{bmatrix} 2 &1 &1 \\ -1&2 &0 \\ 4&2 &4 \end{bmatri
- Determine whether x is an eigenvector of A. A = \begin{bmatrix} 5 & -2\\ -2 & 8 \end{bmatrix} a. x = (2,1) b. x = (1,2) c. x = (1,-2) d. x = (0,-1)
- Suppose the matrix A has eigenvectors u = 1 0 0 , v = 0 3 ? 1 , w = 0 2 ? 2 with corresponding eigenvalues ? = 4 , ? = 4 , ? = 3. Solve x ? = A x with x ( 0 ) = 1 6 ? 2 .
- Find the eigenvalues and corresponding eigenvectors of \begin{bmatrix} 5 & 1\\ 1 & 5 \end{bmatrix}
- Find the eigenvector when the eigenvalue is zero?
- Find eigenvalues and corresponding eigenvectors of the following matrix: \parenthesis 3 1 1 \\ 1 0 2 \\ 1 2 0 \parenthesis (Hint: \lambda = -2 is one of the eigenvalues of the matrix. It would be best
- The Matrix A = begin{pmatrix} -5&1 -4 & -9 end{pmatrix} has an eigen value lambda of multiplicity 2 with corresponding eigenvalue v. Find lambda and v
- Let U denote an n \times n unitary matrix. \text{Show that every (possibly complex) eigenvalue } \lambda \text{ of U satisfies } |\lambda| = 1.
- Find the eigenvalues and corresponding eigenvectors of \begin{bmatrix} 2 & -3\\ -2 & 1 \end{bmatrix} .
- Find the eigenvalues and corresponding eigenvectors of \begin{bmatrix} 0 & 75\\ 0 & 100 \end{bmatrix}
- Consider the matrix A = \begin{bmatrix} 3 & -4\\ 2 & -1 \end{bmatrix} . a. Find the eigenvalues of A . b. Find an eigenvector for each eigenvalue of A . c. Find matrices P, P^{-1}, C such th
- Given that vec v_1 = [ 1 -1 ] and vec v_2 = [ 2 -3 ] are eigenvectors of the matrix A = [ -15 & -12 18& 15 ] determine the corresponding eigenvalues.
Explore our homework questions and answers library
Browse
by subject