If B = `[{:(3,2,0),(2,4,0),(1,1,0):}]`, then what is adjoint of B equal to ?

Options

A. `[{:(0,0,0),(0,0,0),(-2,-1,8):}]`
B. `[{:(0,0,-2),(0,0,-1),(0,0,8):}]`
C. `[{:(0,0,2),(0,0,1),(0,0,0):}]`
D. It does not exist

Explanation

To find the adjoint of the matrix \( B = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 4 & 0 \\ 1 & 1 & 0 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Cofactors of Each Element

The adjoint of a matrix is the transpose of the cofactor matrix. We will calculate the cofactor for each element in the matrix \( B \).

#### Cofactor \( A_{11} \):
- Element: \( 3 \) (1st row, 1st column)
- Remaining matrix after removing 1st row and 1st column: \( \begin{pmatrix} 4 & 0 \\ 1 & 0 \end{pmatrix} \)
- Determinant: \( (4)(0) - (0)(1) = 0 \)
- Cofactor: \( (-1)^{1+1} \cdot 0 = 0 \)

#### Cofactor \( A_{12} \):
- Element: \( 2 \) (1st row, 2nd column)
- Remaining matrix: \( \begin{pmatrix} 2 & 0 \\ 1 & 0 \end{pmatrix} \)
- Determinant: \( (2)(0) - (0)(1) = 0 \)
- Cofactor: \( (-1)^{1+2} \cdot 0 = 0 \)

#### Cofactor \( A_{13} \):
- Element: \( 0 \) (1st row, 3rd column)
- Remaining matrix: \( \begin{pmatrix} 2 & 4 \\ 1 & 1 \end{pmatrix} \)
- Determinant: \( (2)(1) - (4)(1) = 2 - 4 = -2 \)
- Cofactor: \( (-1)^{1+3} \cdot (-2) = -2 \)

#### Cofactor \( A_{21} \):
- Element: \( 2 \) (2nd row, 1st column)
- Remaining matrix: \( \begin{pmatrix} 2 & 0 \\ 1 & 0 \end{pmatrix} \)
- Determinant: \( (2)(0) - (0)(1) = 0 \)
- Cofactor: \( (-1)^{2+1} \cdot 0 = 0 \)

#### Cofactor \( A_{22} \):
- Element: \( 4 \) (2nd row, 2nd column)
- Remaining matrix: \( \begin{pmatrix} 3 & 0 \\ 1 & 0 \end{pmatrix} \)
- Determinant: \( (3)(0) - (0)(1) = 0 \)
- Cofactor: \( (-1)^{2+2} \cdot 0 = 0 \)

#### Cofactor \( A_{23} \):
- Element: \( 0 \) (2nd row, 3rd column)
- Remaining matrix: \( \begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix} \)
- Determinant: \( (3)(1) - (2)(1) = 3 - 2 = 1 \)
- Cofactor: \( (-1)^{2+3} \cdot 1 = -1 \)

#### Cofactor \( A_{31} \):
- Element: \( 1 \) (3rd row, 1st column)
- Remaining matrix: \( \begin{pmatrix} 2 & 0 \\ 4 & 0 \end{pmatrix} \)
- Determinant: \( (2)(0) - (0)(4) = 0 \)
- Cofactor: \( (-1)^{3+1} \cdot 0 = 0 \)

#### Cofactor \( A_{32} \):
- Element: \( 1 \) (3rd row, 2nd column)
- Remaining matrix: \( \begin{pmatrix} 3 & 0 \\ 2 & 0 \end{pmatrix} \)
- Determinant: \( (3)(0) - (0)(2) = 0 \)
- Cofactor: \( (-1)^{3+2} \cdot 0 = 0 \)

#### Cofactor \( A_{33} \):
- Element: \( 0 \) (3rd row, 3rd column)
- Remaining matrix: \( \begin{pmatrix} 3 & 2 \\ 2 & 4 \end{pmatrix} \)
- Determinant: \( (3)(4) - (2)(2) = 12 - 4 = 8 \)
- Cofactor: \( (-1)^{3+3} \cdot 8 = 8 \)

### Step 2: Form the Cofactor Matrix

The cofactor matrix is:
\[ \text{Cofactor Matrix} = \begin{pmatrix} 0 & 0 & -2 \\ 0 & 0 & -1 \\ 0 & 0 & 8 \end{pmatrix} \]

### Step 3: Transpose the Cofactor Matrix

The adjoint of matrix \( B \) is the transpose of the cofactor matrix:
\[ \text{Adjoint}(B) = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ -2 & -1 & 8 \end{pmatrix} \]

### Final Answer
Thus, the adjoint of matrix \( B \) is:
\[ \text{Adjoint}(B) = \begin{pmatrix} 0 & 0 & -2 \\ 0 & 0 & -1 \\ 0 & 0 & 8 \end{pmatrix} \]