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GATE 2026 DA – Question 46

Linear Algebra · Eigenvalues, eigenvectors and determinant · 2 marks · Multiple choice

Let $\gamma_1, \gamma_2, \gamma_3$ be the eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix}$, where $t \in [-\pi, \pi]$ is in radians.

Which one of the following options lists all the possible values of $t$ satisfying $\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}$?

  1. $\left\{\frac{\pi}{3}, -\frac{\pi}{4}\right\}$
  2. $\left\{\frac{\pi}{4}, -\frac{\pi}{3}\right\}$
  3. $\left\{\frac{\pi}{4}, -\frac{\pi}{4}\right\}$
  4. $\left\{\frac{\pi}{3}, -\frac{\pi}{3}\right\}$

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Correct answer: (C) $\left\{\frac{\pi}{4}, -\frac{\pi}{4}\right\}$

Explanation

The matrix is block diagonal, with the eigenvalue 1 and the rotation block, whose eigenvalues are $\cos t \pm i\sin t$. The sum of the eigenvalues is the trace, $1 + 2\cos t$. Setting $1 + 2\cos t = 1 + \sqrt{2}$ gives $\cos t = \frac{\sqrt{2}}{2}$, so $t = \pm\frac{\pi}{4}$ in $[-\pi, \pi]$.