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GATE 2022 IN – Question 49

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

Matrix $A=\begin{bmatrix}4&3\\9&-2\end{bmatrix}$ has eigenvalues −5 and 7. Select its eigenvector(s).

  1. $\begin{bmatrix}1\\1\end{bmatrix}$
  2. $\begin{bmatrix}3\\4\end{bmatrix}$
  3. $\begin{bmatrix}2\\-6\end{bmatrix}$
  4. $\begin{bmatrix}2\\8\end{bmatrix}$

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Correct answer: (A) $\begin{bmatrix}1\\1\end{bmatrix}$; (C) $\begin{bmatrix}2\\-6\end{bmatrix}$

Explanation

For A=[[4,3],[9,−2]], multiplying (1,1) gives (7,7), so A is a λ=7 eigenvector. Multiplying (2,−6) gives (−10,30), a λ=−5 multiple; this is the vector shown in option C.

Thus **A,C**.