To solve the problem, we need to calculate the determinant of the matrix \( A \) and find its range when \( \theta \) lies in the interval \( \left(\frac{3\pi}{4}, \frac{5\pi}{4}\right) \).
### Step 1: Write down the matrix \( A \)
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
1 & \sin \theta & 1 \\
-\sin \theta & 1 & \sin \theta \\
-1 & -\sin \theta & 1
\end{bmatrix}
\]
### Step 2: Calculate the determinant of \( A \)
To find the determinant of \( A \), we can use the formula for the determinant of a \( 3 \times 3 \) matrix:
\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
where the matrix is represented as:
\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]
For our matrix:
- \( a = 1, b = \sin \theta, c = 1 \)
- \( d = -\sin \theta, e = 1, f = \sin \theta \)
- \( g = -1, h = -\sin \theta, i = 1 \)
Calculating the determinant:
\[
\text{det}(A) = 1(1 \cdot 1 - \sin \theta \cdot (-\sin \theta)) - \sin \theta(-\sin \theta \cdot 1 - \sin \theta \cdot (-1)) + 1(-\sin \theta \cdot (-\sin \theta) - 1 \cdot 1)
\]
This simplifies to:
\[
= 1(1 + \sin^2 \theta) + \sin \theta(\sin \theta + \sin \theta) + 1(\sin^2 \theta - 1)
\]
\[
= 1 + \sin^2 \theta + 2\sin^2 \theta + \sin^2 \theta - 1
\]
\[
= 2 + 2\sin^2 \theta
\]
### Step 3: Determine the range of \( \sin^2 \theta \)
Next, we need to find the range of \( \sin^2 \theta \) when \( \theta \) is in the interval \( \left(\frac{3\pi}{4}, \frac{5\pi}{4}\right) \).
- At \( \theta = \frac{3\pi}{4} \), \( \sin \theta = \frac{1}{\sqrt{2}} \) so \( \sin^2 \theta = \frac{1}{2} \).
- At \( \theta = \frac{5\pi}{4} \), \( \sin \theta = -\frac{1}{\sqrt{2}} \) so \( \sin^2 \theta = \frac{1}{2} \).
As \( \theta \) varies from \( \frac{3\pi}{4} \) to \( \frac{5\pi}{4} \), \( \sin \theta \) decreases from \( \frac{1}{\sqrt{2}} \) to \( -\frac{1}{\sqrt{2}} \), thus \( \sin^2 \theta \) varies from \( 0 \) to \( \frac{1}{2} \).
### Step 4: Find the range of \( 2 + 2\sin^2 \theta \)
Now, substituting the range of \( \sin^2 \theta \) into the expression for the determinant:
\[
\text{det}(A) = 2 + 2\sin^2 \theta
\]
When \( \sin^2 \theta \) varies from \( 0 \) to \( \frac{1}{2} \):
- Minimum value: \( 2 + 2 \cdot 0 = 2 \)
- Maximum value: \( 2 + 2 \cdot \frac{1}{2} = 3 \)
### Conclusion
Thus, the range of the determinant \( \text{det}(A) \) when \( \theta \) lies in the interval \( \left(\frac{3\pi}{4}, \frac{5\pi}{4}\right) \) is:
\[
\text{det}(A) \in [2, 3]
\]
