Final answer:
This answer provides explanations for the given statements about eigenvalues and eigenvectors
Step-by-step explanation:
a. A matrix A is not invertible if and only if 0 is an eigenvalue of A. This means that if A is invertible, then 0 is not an eigenvalue of A and vice versa.
b. If Ax = λx for some vector x, then λ is an eigenvalue of A. This is because λ represents the scalar by which A stretches or shrinks the vector x.
c. A number λ is an eigenvalue of A if and only if (A - λI)x = 0 has a nontrivial solution, where I is the identity matrix. In other words, λ is an eigenvalue if there is a non-zero vector x such that (A - λI)x = 0.
d. An eigenspace of A is the set of all eigenvectors corresponding to a specific eigenvalue of A. It is not necessarily equal to the kernel of a certain matrix.
e. If Ax = λx for some vector x, then x is an eigenvector of A corresponding to the eigenvalue λ.
f. The eigenvalues of a matrix are not always on its main diagonal. While diagonal matrices have eigenvalues on the main diagonal, other matrices can have eigenvalues located elsewhere.
g. If one multiple of one row of A is added to another row, the eigenvalues of A do not change. This operation does not affect the eigenvalues of the matrix.
h. If λ is a factor of the characteristic polynomial of A, then λ is an eigenvalue of A. The characteristic polynomial is formed by setting the determinant of (A - λI) equal to zero.
i. If x and y are linearly independent eigenvectors of A with distinct eigenvalues, then they correspond to distinct eigenvalues. The eigenvectors associated with distinct eigenvalues are always linearly independent.