Question

Let y1, y2, y3 eigen value of M = $\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & \sin t & \cos t \end{bmatrix}$ Where t $\in$ [-$\pi$,$\pi$] \& y$_1$ + y$_2$ + y$_3$ = 1 + $\sqrt{2}$ then t = ?

• A. $\frac{\pi}{4}, \frac{\pi}{3}$
• B. $\frac{\pi}{3}, -\frac{\pi}{3}$
• C. $-\frac{\pi}{3}, \frac{\pi}{4}$
• D. $\frac{\pi}{4}, -\frac{\pi}{4}$

Hint:

For questions involving the sum of eigenvalues, always check if you can use the trace property. It's a significant shortcut that avoids the need to calculate the characteristic polynomial and solve for the eigenvalues directly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a 3x3 matrix M and the sum of its eigenvalues. We need to find the possible values of the parameter 't' within a given interval.
Step 2: Key Formula or Approach:
A fundamental property of matrices states that the sum of the eigenvalues of a matrix is equal to the trace of the matrix. The trace of a square matrix is the sum of the elements on its main diagonal. \[ \sum_{i=1}^{n} \lambda_i = \text{Tr}(M) \] Step 3: Detailed Explanation:
First, let's find the trace of the given matrix M: \[ M = \begin{bmatrix} 1 & 0 & 0
0 & \cos t & \sin t
0 & \sin t & \cos t \end{bmatrix} \] The elements on the main diagonal are 1, cos(t), and cos(t). \[ \text{Tr}(M) = 1 + \cos(t) + \cos(t) = 1 + 2\cos(t) \] We are given that the sum of the eigenvalues is $1 + \sqrt{2}$. \[ y_1 + y_2 + y_3 = 1 + \sqrt{2} \] According to the property, we can equate the trace and the sum of eigenvalues: \[ 1 + 2\cos(t) = 1 + \sqrt{2} \] Now, we solve for t: \[ 2\cos(t) = \sqrt{2} \] \[ \cos(t) = \frac{\sqrt{2}}{2} \] We need to find the values of t in the interval $[-\pi, \pi]$ that satisfy this equation. The principal value for $t$ is $\frac{\pi}{4}$ (or 45$^\circ$). Since the cosine function is an even function ($\cos(-t) = \cos(t)$), the value in the negative part of the interval is $t = -\frac{\pi}{4}$. Both $\frac{\pi}{4}$ and $-\frac{\pi}{4}$ are within the specified interval $[-\pi, \pi]$.
Step 4: Final Answer:
The possible values for t are $\frac{\pi}{4}$ and $-\frac{\pi}{4}$.