Question

Ais an nx n matrix. Mark each statement below True or False. Justify each answer

a. If Ax = λχ for some scalar λ, then x is an eigenvector of A. Choose the correct answer below
A. False. The equation Ax-1x is not used to determine eigenvectors. If 1Ax-0 for some scalar λ, then x is an eigenvector of A.
B. True. If Ax-2x for some scalar λ, then x is an eigenvector of A because the only solution to this equation is the trivial solution.
C. False. The condition that Ax-1x for some scalar λ is not sufficient to determine if x is an eigenvector of A. The vector x must be nonzero
D. True. If Ax-λχ for some scalar λ, then x is an eigenvector of A because λ is an inverse of A.
b. If v, and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. Choose the correct answer belovw .
A. True. If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because all eigenvectors of a single eigenvalue are linearly dependent.
B. False. There m ay be linearly independent eigenvectors that both correspond to the same eigenvalue
C. True. If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because eigenvectors must be nonzero
D. False. Every eigenvalue has an infinite number of corresponding eigenvectors
c. A steady-state vector for a stochastic matrix is actually an eigenvector. Choose the correct answer below.
A. False. A steady-state vector for a stochastic matrix is not an eigenvector because it does not satisfy the equation Ax= 0
B. True. A steady-state vector for a stochastic matrix is actually an eigenvector because it satisfies the equation Ax = 0
C. False. A steady-state vector for a stochastic matrix is not an eigenvector because it does not satisfy the equation Ax x.
D. True. A steady-state vector for a stochastic matrix is actually an eigenvector because it satisfies the equation Ax= x

Answer 1

If Ax-2x for some scalar λ, then x is an eigenvector of A because the only solution to this equation is the trivial solution.

Explanation:

The eigen vector is linear transformation which is a non zero vector which changes due to scalar factor. The eigen factor value is denoted by lambda. The eigen vector is a latent vector.