The GATE Grind

GATE 2018 CS – Question 36

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

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?

  1. Only I and III are necessarily true
  2. Only II is necessarily true
  3. Only I and II are necessarily true
  4. Only II and III are necessarily true

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Correct answer: (D) Only II and III are necessarily true

Explanation

$P$ is $2\times2$ with a single independent eigenvector, so it is defective: it has a repeated eigenvalue (II) and cannot be diagonalized (III). The repeated eigenvalue can be non-zero, so $P$ may still be invertible (I is not necessary).