`R=[(x,0,0),(0,y,0),(0,0,z)],x/sintheta=y/sin(0+(2pi)/3)=z/sin(0+(4pi)/3)`
Statement-1: Trace (R) = 0
Statement-2: Trace (adj(adj(R))=0

Options

A. Statement-1 is true and statement-2 is false
B. Statement-1 is false and statement-2 is false
C. Statement-1 is false and statement-2 is true
D. Statement-1 is true and statement-2 is true

Answer

To solve the problem step-by-step, we will analyze the given matrix \( R \) and the conditions provided in the question.

### Step 1: Define the Matrix \( R \)
The matrix \( R \) is given as:
\[ R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix} \]

### Step 2: Analyze the Ratios
We have the ratios:
\[ \frac{x}{\sin \theta} = \frac{y}{\sin\left(0 + \frac{2\pi}{3}\right)} = \frac{z}{\sin\left(0 + \frac{4\pi}{3}\right)} \]
Let’s denote this common ratio as \( k \):
\[ x = k \sin \theta, \quad y = k \sin\left(\frac{2\pi}{3}\right), \quad z = k \sin\left(\frac{4\pi}{3}\right) \]

### Step 3: Calculate \( y \) and \( z \)
From trigonometric values:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(180^\circ - 60^\circ\right) = \sin 60^\circ = \frac{\sqrt{3}}{2} \]
\[ \sin\left(\frac{4\pi}{3}\right) = \sin\left(180^\circ + 60^\circ\right) = -\sin 60^\circ = -\frac{\sqrt{3}}{2} \]
Thus, we can express \( y \) and \( z \) as:
\[ y = k \cdot \frac{\sqrt{3}}{2}, \quad z = k \cdot \left(-\frac{\sqrt{3}}{2}\right) \]

### Step 4: Calculate \( x + y + z \)
Now, we can find \( x + y + z \):
\[ x + y + z = k \sin \theta + k \cdot \frac{\sqrt{3}}{2} - k \cdot \frac{\sqrt{3}}{2} = k \sin \theta \]
This implies:
\[ x + y + z = k \sin \theta \]

### Step 5: Determine the Trace of \( R \)
The trace of matrix \( R \) is given by:
\[ \text{Trace}(R) = x + y + z = k \sin \theta \]
To satisfy Statement 1, we need \( \text{Trace}(R) = 0 \). This occurs if:
\[ k \sin \theta = 0 \]
Thus, either \( k = 0 \) or \( \sin \theta = 0 \).

### Step 6: Calculate the Adjoint of \( R \)
The adjoint of \( R \) is given by:
\[ \text{adj}(R) = \begin{pmatrix} yz & 0 & 0 \\ 0 & xz & 0 \\ 0 & 0 & xy \end{pmatrix} \]

### Step 7: Calculate the Adjoint of the Adjoint
Now we need to find \( \text{adj}(\text{adj}(R)) \):
\[ \text{adj}(\text{adj}(R)) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & xyz \end{pmatrix} \]
The trace of this matrix is \( 0 \) if \( x + y + z = 0 \).

### Conclusion
Both statements are correct:
- Statement 1: \( \text{Trace}(R) = 0 \)
- Statement 2: \( \text{Trace}(\text{adj}(\text{adj}(R))) = 0 \)