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GATE | GATE CS 2018 | Question 63

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Consider a matrix P whose only eigenvectors are the multiples of \begin{bmatrix} 1\\  4 \end{bmatrix}. 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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