If `A=[(costheta,-sintheta,0),(sintheta,costheta,0),(0,0,1)]`, then `adj A` equals to

Options

A. A
B. I
C. O
D. `A^(2)`

Answer

To find the adjoint of the matrix \( A = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Find the Cofactor Matrix
The cofactor matrix is obtained by calculating the cofactors for each element of the matrix \( A \).

1. **Cofactor \( C_{11} \)**:
- Minor of \( A_{11} \) (remove first row and first column):
\[ \text{Minor} = \begin{vmatrix} \cos \theta & 0 \\ 0 & 1 \end{vmatrix} = \cos \theta \]
- Cofactor \( C_{11} = (-1)^{1+1} \cdot \text{Minor} = \cos \theta \)

2. **Cofactor \( C_{12} \)**:
- Minor of \( A_{12} \) (remove first row and second column):
\[ \text{Minor} = \begin{vmatrix} \sin \theta & 0 \\ 0 & 1 \end{vmatrix} = \sin \theta \]
- Cofactor \( C_{12} = (-1)^{1+2} \cdot \text{Minor} = -\sin \theta \)

3. **Cofactor \( C_{13} \)**:
- Minor of \( A_{13} \) (remove first row and third column):
\[ \text{Minor} = \begin{vmatrix} \sin \theta & \cos \theta \\ 0 & 0 \end{vmatrix} = 0 \]
- Cofactor \( C_{13} = (-1)^{1+3} \cdot \text{Minor} = 0 \)

4. **Cofactor \( C_{21} \)**:
- Minor of \( A_{21} \) (remove second row and first column):
\[ \text{Minor} = \begin{vmatrix} -\sin \theta & 0 \\ 0 & 1 \end{vmatrix} = -\sin \theta \]
- Cofactor \( C_{21} = (-1)^{2+1} \cdot \text{Minor} = \sin \theta \)

5. **Cofactor \( C_{22} \)**:
- Minor of \( A_{22} \) (remove second row and second column):
\[ \text{Minor} = \begin{vmatrix} \cos \theta & 0 \\ 0 & 1 \end{vmatrix} = \cos \theta \]
- Cofactor \( C_{22} = (-1)^{2+2} \cdot \text{Minor} = \cos \theta \)

6. **Cofactor \( C_{23} \)**:
- Minor of \( A_{23} \) (remove second row and third column):
\[ \text{Minor} = \begin{vmatrix} \cos \theta & -\sin \theta \\ 0 & 0 \end{vmatrix} = 0 \]
- Cofactor \( C_{23} = (-1)^{2+3} \cdot \text{Minor} = 0 \)

7. **Cofactor \( C_{31} \)**:
- Minor of \( A_{31} \) (remove third row and first column):
\[ \text{Minor} = \begin{vmatrix} -\sin \theta & 0 \\ \cos \theta & 0 \end{vmatrix} = 0 \]
- Cofactor \( C_{31} = (-1)^{3+1} \cdot \text{Minor} = 0 \)

8. **Cofactor \( C_{32} \)**:
- Minor of \( A_{32} \) (remove third row and second column):
\[ \text{Minor} = \begin{vmatrix} \cos \theta & 0 \\ \sin \theta & 0 \end{vmatrix} = 0 \]
- Cofactor \( C_{32} = (-1)^{3+2} \cdot \text{Minor} = 0 \)

9. **Cofactor \( C_{33} \)**:
- Minor of \( A_{33} \) (remove third row and third column):
\[ \text{Minor} = \begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix} = \cos^2 \theta + \sin^2 \theta = 1 \]
- Cofactor \( C_{33} = (-1)^{3+3} \cdot \text{Minor} = 1 \)

### Step 2: Construct the Cofactor Matrix
The cofactor matrix is:
\[ C = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 3: Transpose the Cofactor Matrix
The adjoint of matrix \( A \) is the transpose of the cofactor matrix:
\[ \text{adj} A = C^T = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Final Answer
Thus, the adjoint of matrix \( A \) is:
\[ \text{adj} A = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]