GATE 2026 BM – Question 58

Engineering Mathematics · Linear Algebra · 2 marks · Numerical answer

Vector $(2,a)^T$ is an eigenvector for the smallest eigenvalue of $\begin{pmatrix}1&2\\2&1\end{pmatrix}$. Find a as an integer.

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Correct answer: -2 to -2

Explanation

The characteristic roots are three and minus one. For λ=−1, the first row of Av=λv gives 2+2a=−2, so a=−2. The second row then gives 4−2=2=−a, confirming the same eigenvector. Thus a is minus two.

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