GATE 2025 IN – Question 32
One eigenvector of $A=\begin{bmatrix}-1&-1\\x&-4\end{bmatrix}$ is along $\begin{bmatrix}\alpha\\2\alpha\end{bmatrix}$, where $\alpha$ is any nonzero real number. The value of $x$ is ______ (in integer).
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Correct answer: 2 to 2
Explanation
The eigenvector scale cancels, so use $v=(1,2)^T$.
The first row of $Av=\lambda v$ gives $\lambda=-1-2=-3$.
The second row gives $x-8=2\lambda=-6$, so **$x=2$**.