Find the cofactors of elements `a_(12),a_(22),a_(32)` respectively of the matrix `[(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,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 given matrix

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

we will follow these steps:

### Step 1: Understand the Cofactor Definition
The cofactor \( C_{ij} \) of an element \( a_{ij} \) is defined as:

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

where \( M_{ij} \) is the minor of the element \( a_{ij} \), which is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column from the original matrix.

### Step 2: Calculate \( C_{12} \)
1. **Identify \( a_{12} \)**: The element in the first row and second column is \( \sin \theta \).
2. **Find the minor \( M_{12} \)**: Remove the first row and second column:

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

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 3: Calculate \( C_{22} \)
1. **Identify \( a_{22} \)**: The element in the second row and second column is \( 1 \).
2. **Find the minor \( M_{22} \)**: Remove the second row and second column:

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

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 4: Calculate \( C_{32} \)
1. **Identify \( a_{32} \)**: The element in the third row and second column is \( -\sin \theta \).
2. **Find the minor \( M_{32} \)**: Remove the third row and second column:

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

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
The cofactors are:
- \( C_{12} = 0 \)
- \( C_{22} = 2 \)
- \( C_{32} = -2\sin \theta \)

### Summary of Cofactors
Thus, the cofactors of the elements \( a_{12}, a_{22}, a_{32} \) are \( 0, 2, -2\sin \theta \) respectively.