Find the cofactors of elements `a_12,a_22,a_32` respectively of the matrix `[(1,sin theta ,1),(-sin theta ,1, sin theta),(-1, -sin theta ,1)]`:

Options

A. `0,2,-2 sin theta`
B. `2,0, 2 sin theta`
C. `2,0 -2 sin theta`
D. `-2 sin theta ,2,0`

Answer

To find the cofactors of the elements \( a_{12}, a_{22}, a_{32} \) of the matrix

\[ A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}, \]

we will follow the definition of the cofactor, which is given by:

\[ C_{ij} = (-1)^{i+j} M_{ij}, \]

where \( M_{ij} \) is the minor of the element \( a_{ij} \).

### Step 1: Calculate the cofactor \( C_{12} \)

1. **Identify the element \( a_{12} \)**: This is the element in the first row and second column, which is \( \sin \theta \).
2. **Find the minor \( M_{12} \)**: To find \( M_{12} \), we remove the first row and the second column from the matrix \( A \):

\[ M_{12} = \begin{vmatrix} -\sin \theta & \sin \theta \\ -1 & 1 \end{vmatrix} \]

3. **Calculate the determinant**:

\[ M_{12} = (-\sin \theta)(1) - (\sin \theta)(-1) = -\sin \theta + \sin \theta = 0. \]

4. **Calculate the cofactor**:

\[ C_{12} = (-1)^{1+2} M_{12} = (-1)^{3} \cdot 0 = 0. \]

### Step 2: Calculate the cofactor \( C_{22} \)

1. **Identify the element \( a_{22} \)**: This is the element in the second row and second column, which is \( 1 \).
2. **Find the minor \( M_{22} \)**: To find \( M_{22} \), we remove the second row and the second column from the matrix \( A \):

\[ M_{22} = \begin{vmatrix} 1 & 1 \\ -1 & 1 \end{vmatrix} \]

3. **Calculate the determinant**:

\[ M_{22} = (1)(1) - (1)(-1) = 1 + 1 = 2. \]

4. **Calculate the cofactor**:

\[ C_{22} = (-1)^{2+2} M_{22} = 1 \cdot 2 = 2. \]

### Step 3: Calculate the cofactor \( C_{32} \)

1. **Identify the element \( a_{32} \)**: This is the element in the third row and second column, which is \( -\sin \theta \).
2. **Find the minor \( M_{32} \)**: To find \( M_{32} \), we remove the third row and the second column from the matrix \( A \):

\[ M_{32} = \begin{vmatrix} 1 & 1 \\ -\sin \theta & \sin \theta \end{vmatrix} \]

3. **Calculate the determinant**:

\[ M_{32} = (1)(\sin \theta) - (1)(-\sin \theta) = \sin \theta + \sin \theta = 2 \sin \theta. \]

4. **Calculate the cofactor**:

\[ C_{32} = (-1)^{3+2} M_{32} = (-1)^{5} \cdot (2 \sin \theta) = -2 \sin \theta. \]

### Final Results

Thus, the cofactors are:

- \( C_{12} = 0 \)
- \( C_{22} = 2 \)
- \( C_{32} = -2 \sin \theta \)