Question

Consider the following two statements.

Statement A: If two matrices are similar, then they have the same eigenvalues.
Statement B: If two matrices have the same eigenvalues, then the two matrices are similar.
a. They are both false
b. They are both true
c. A is true, B is false
d. A is false, B is true

Answer 1

The correct answer is d. A is false, B is true.

Step-by-step explanation:

The correct answer is d. A is false, B is true.

Statement A: If two matrices are similar, then they have the same eigenvalues. This statement is false. Similar matrices do not necessarily have the same eigenvalues. Eigenvalues are the values that satisfy the equation Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvalue. Similar matrices have the same eigenvectors, but their eigenvalues can be different.

Statement B: If two matrices have the same eigenvalues, then the two matrices are similar. This statement is true. Matrices with the same eigenvalues can be similar. If two matrices have the same eigenvalues, it means that they have the same characteristic equation, which implies that they have the same eigenvectors. Therefore, the matrices are similar.