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GATE 2024 MT – Question 36

Engineering Mathematics · Linear algebra and matrices · 2 marks · Multiple choice

If $\begin{bmatrix}1&2\\8&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\lambda\begin{bmatrix}x\\y\end{bmatrix}$, where $x,y$ are not both zero, the values of $\lambda$ are

  1. 5, −3
  2. 4, −4
  3. 3, −5
  4. 5, −4

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Correct answer: (A) 5, −3

Explanation

Nonzero eigenvectors require det(A−λI)=0.

The determinant is (1−λ)²−16, so 1−λ=±4.

Therefore the eigenvalues are **5 and −3, A**.