Question

Let \[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \text{ be a non-zero } 3 \times 3 \text{ matrix, where} \]

\[ x = \sin \theta, \quad y = \sin \left( \theta + \frac{2\pi}{3} \right), \quad z = \sin \left( \theta + \frac{4\pi}{3} \right) \]

and \( \theta \neq 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). For a square matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal entries of \( M \). Then, among the statements:

1. \(\text{Trace}(R) = 0\)
2. If \(\text{trace}(\text{adj}(\text{adj}(R))) = 0\), then \( R \) has exactly one non-zero entry.

Which of the following is true?

• A. Only(I) is true
• B. Only (II) is true
• C. Neither (I) nor (II) is true
• D. Both (I) and (II) are true

Answer 1

The Correct Option is A

Calculate the trace of \( R \): Since \( x + y + z = \sin \theta + \sin \left( \theta + \frac{2\pi}{3} \right) + \sin \left( \theta + \frac{4\pi}{3} \right) = 0 \), we have:

\[ \text{trace}(R) = x + y + z = 0. \]

Thus, statement (I) is true.

Examine statement (II): \(\text{adj}(R) = \begin{pmatrix} yz & 0 & 0 \\ 0 & xz & 0 \\ 0 & 0 & xy \end{pmatrix}\). Therefore,

\[ \text{adj}(\text{adj}(R)) = \begin{pmatrix} x^2yz & 0 & 0 \\ 0 & xy^2z & 0 \\ 0 & 0 & xyz^2 \end{pmatrix}. \]

The trace of \(\text{adj}(\text{adj}(R))\) is \( xyz(x + y + z) = 0 \), even if \( R \) has more than one non-zero entry.

Thus, statement (II) is false.