GATE 2025 BT – Question 63
Let $A = \begin{pmatrix}1 & 0 & 1\\0 & k & 0\\3 & 0 & -1\end{pmatrix}$. If the eigenvalues of $A$ are $-2$, 1, and 2, then the value of $k$ is ______________. (Answer in integer)
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Correct answer: 1
Explanation
The middle row and column are only $(0, k, 0)$, so $k$ is itself an eigenvalue. The remaining block $\begin{pmatrix}1 & 1\\3 & -1\end{pmatrix}$ has the characteristic equation $\lambda^2 - 4 = 0$, so its eigenvalues are $\pm 2$. The given eigenvalues are $-2, 1, 2$, so $k = 1$.