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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Last Updated :
09 Mar, 2018
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