If $A=\begin{bmatrix}1&sin~ heta&1\ -sin~ heta&1&sin~ heta\ -1&-sin~ heta&1\end{bmatrix},$ where $0\le heta\le2\pi,$ then :

Question

If $A=\begin{bmatrix}1&sin~	heta&1\ -sin~	heta&1&sin~	heta\ -1&-sin~	heta&1\end{bmatrix},$ where $0\le	heta\le2\pi,$ then :

Accepted Answer

To determine the correct option for the determinant of the matrix $ A = \begin{bmatrix} 1 & \sin \theta & 1 \ -\sin \theta & 1 & \sin \theta \ -1 & -\sin \theta & 1 \end{bmatrix} $, we need to calculate the determinant and analyze its possible values over the given range of $ \theta $.

### Calculation of the Determinant

The determinant of a 3x3 matrix $ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} $ is given by:

$$
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
$$

Applying this formula to our matrix $ A $:

$$
\det(A) = 1(1 \cdot 1 - \sin \theta \cdot (-\sin \theta)) - \sin \theta(-\sin \theta \cdot 1 - \sin \theta \cdot (-1)) + 1(-\sin \theta \cdot (-\sin \theta) - 1 \cdot (-1))
$$

Simplifying each term:

1. $ 1(1 + \sin^2 \theta) = 1 + \sin^2 \theta $
2. $ -\sin \theta(\sin \theta + \sin \theta) = -2\sin^2 \theta $
3. $ 1(\sin^2 \theta + 1) = \sin^2 \theta + 1 $

Adding these results together:

$$
\det(A) = (1 + \sin^2 \theta) - 2\sin^2 \theta + (\sin^2 \theta + 1) = 2
$$

### Analysis of the Determinant

The determinant simplifies to a constant value of 2, regardless of the value of $ \theta $ within the range $ 0 \leq \theta \leq 2\pi $.

### Justification for Option 4

- **Option 4: $ \det A \in [2, 4] $** is correct because the determinant is exactly 2 for all $ \theta $, which falls within the closed interval $[2, 4]$.

### Explanation of Why Other Options are Incorrect

- **Option 1: $ \det A = 0 $** is incorrect because the determinant is 2, not 0.
- **Option 2: $ \det A \in (2, \infty) $** is incorrect because the determinant is exactly 2, not greater than 2.
- **Option 3: $ \det A \in (2, 4) $** is incorrect because the determinant is exactly 2, which is not within the open interval $(2, 4)$.

Thus, the correct answer is Option 4, as it accurately describes the value of the determinant over the specified range of $ \theta $.