Consider a matrix P whose only eigenvectors are the multiples of .
Consider the following statements.
(I) P does not have an inverse
(II) P has a repeated eigenvalue
(III) P cannot be diagonalized
Which one of the following options is correct?
(A) Only I and III are necessarily true
(B) Only II is necessarily true
(C) Only I and II are necessarily true
(D) Only II and III are necessarily true
Explanation: Repeated eigenvectors come from repeated eigenvalues. Therefore, statement (I) may not be correct, take any Identity matrix which has same eigenvalues but determinant so inverse exists.
A matrix with repeated eigenvalues can or can not be diagonalized, repeated eigenvalues are necessary but not sufficient for a matrix to not be diagonalizable. But if all eigenvalues are distinct then we can be sure to be able to diagonalize it.
In other words: as soon as all eigenvalues are distinct then we can be sure to be able to diagonalize it.
Therefore, only statements (II) and (III) are necessarily true.