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GATE 2021 CE (CE2) – Question 37

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple choice

Find the smallest eigenvalue and matching eigenvector of $\begin{bmatrix}2&-2\\-1&6\end{bmatrix}$.

  1. $1.55,\ (2.00,0.45)^T$
  2. $2.00,\ (1.00,1.00)^T$
  3. $1.55,\ (-2.55,-0.45)^T$
  4. $1.55,\ (2.00,-0.45)^T$

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Correct answer: (A) $1.55,\ (2.00,0.45)^T$

Explanation

The characteristic polynomial is $(2-\lambda)(6-\lambda)-2=\lambda^2-8\lambda+10$. Thus the smaller eigenvalue is $4-\sqrt6=1.55051$. Its first row gives $v_2/v_1=(2-\lambda)/2=0.224745$, matching (2,0.45). **Answer: A.**