GATE 2026 DA Q.46: solution and answer

GATE 2026 · DAQ.46 · 2 marks · MCQ

Let $γ_{1}, γ_{2}, γ_{3}$ be the eigenvalues of the matrix $100 0 cos t - sin t 0 sin t cos t$ , where $t \in [- π, π]$ is in radians.

Which one of the following options lists all the possible values of $t$ satisfying $γ_{1} + γ_{2} + γ_{3} = 1 + 2$ ?

(A)
${\frac{π}{3}, - \frac{π}{4}}$
(B)
${\frac{π}{4}, - \frac{π}{3}}$
(C)
${\frac{π}{4}, - \frac{π}{4}}$
(D)
${\frac{π}{3}, - \frac{π}{3}}$

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Answer and solution

ANSWER

The set of all possible values is $t \in {\frac{π}{4}, - \frac{π}{4}}$ , which is option (C).

Official answer: CSource: Official answer key, GATE 2026 DA (IIT Guwahati)

The sum of the eigenvalues of a square matrix equals its trace, whether the eigenvalues are real or complex: $γ_{1} + γ_{2} + γ_{3} = tr (A)$ .
The diagonal entries are $1$ , $cos t$ and $cos t$ , so $tr (A) = 1 + 2 cos t .$
Set the trace equal to the given sum: $1 + 2 cos t = 1 + 2$ . This gives $cos t = \frac{2}{2}$ .
On $[- π, π]$ , $cos t = \frac{2}{2}$ holds exactly at $t = \frac{π}{4}$ and $t = - \frac{π}{4}$ . Cosine is even, so the two values are symmetric. No other angle in this interval has this cosine: $cos t$ is positive only for $∣ t ∣ < \frac{π}{2}$ , and on $[0, \frac{π}{2})$ it is strictly decreasing, so it takes the value $\frac{2}{2}$ once.
The full set of values is ${\frac{π}{4}, - \frac{π}{4}}$ , which is option (C).

KEY IDEA

The sum of the eigenvalues is the trace, so there is no need to find the eigenvalues. Here the trace is $1 + 2 cos t$ . Solve $cos t = \frac{2}{2}$ on $[- π, π]$ , where both $t$ and $- t$ qualify because cosine is even.

WATCH OUT FOR

Computing the characteristic polynomial and the complex eigenvalues $cos t \pm i sin t$ takes longer and invites sign errors. Another slip is to give only the positive angle, or to pair $\frac{π}{4}$ with $- \frac{π}{3}$ because $sin t$ also appears in the matrix. The trace involves only $cos t$ , so $t$ and $- t$ always give the same sum.

From the GATE 2026 DA paper. Every question of this paper · Download the full paper · All 6 Eigenvalues questions, with notes

GATE 2026 DA Q.46: solution and answer

Answer and solution

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