Question

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 \(\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}\)?

• A. \(\{\pi/3, -\pi/4\}\)
• B. \(\{\pi/4, -\pi/3\}\)
• C. \(\{\pi/4, -\pi/4\}\)
• D. \(\{\pi/3, -\pi/3\}\)

Hint:

The sum of eigenvalues equals the matrix trace; set \(1+2\cos t = 1+\sqrt2\) and solve for \(\cos t\), then find all matching angles in \([-\pi,\pi]\).

Answer 1

The Correct Option is C

Step 1: Recall how the sum of eigenvalues relates to the matrix.
For any square matrix, the sum of its eigenvalues always equals the trace of the matrix, that is, the sum of the entries on the main diagonal. This holds regardless of whether the eigenvalues are real or complex.

Step 2: Compute the trace of the given matrix.
The diagonal entries are \(1\), \(\cos t\) and \(\cos t\), so
\[ \gamma_1 + \gamma_2 + \gamma_3 = \text{trace} = 1 + \cos t + \cos t = 1 + 2\cos t \]

Step 3: Set the trace equal to the given value and solve for \(\cos t\).
\[ 1 + 2\cos t = 1 + \sqrt{2} \] \[ 2\cos t = \sqrt{2} \] \[ \cos t = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \]

Step 4: Find all \(t\) in \([-\pi, \pi]\) with this cosine value.
\(\cos t = 1/\sqrt{2}\) corresponds to a reference angle of \(\pi/4\) (45 degrees), and cosine is positive in both the first and fourth quadrants. Within \([-\pi, \pi]\), this gives
\[ t = \frac{\pi}{4} \quad \text{or} \quad t = -\frac{\pi}{4} \]

Step 5: Rule out the other options.
\(t=\pi/3\) gives \(\cos t = 1/2\), which does not satisfy \(\cos t = 1/\sqrt{2}\), so any option mixing in \(\pm\pi/3\) is wrong. Only the pair \(\{\pi/4, -\pi/4\}\) satisfies the equation.

Final Answer:
The correct option is (C).\[ \boxed{t \in \{\pi/4, -\pi/4\}} \]