GATE | GATE CS 2018 | Question 63

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


Answer: (D)

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.

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