If `{:S=[(a,b),(c,d)]:}`, then adj S is equal to

Options

A. `{:[(-d,-b),(-c,a)]:}`
B. `{:[(d,-b),(-c,a)]:}`
C. `{:[(d,b),(c,a)]:}`
D. `{:[(d,c),(b,a)]:}`

Explanation

To find the adjoint of the matrix \( S = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), we will follow the properties of the adjoint of a 2x2 matrix.

### Step-by-Step Solution:

1. **Identify the Matrix**:
We have the matrix \( S \) defined as:
\[ S = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

2. **Recall the Property of Adjoint**:
For a 2x2 matrix \( \begin{pmatrix} p & q \\ r & s \end{pmatrix} \), the adjoint (or adjugate) is given by:
\[ \text{adj} = \begin{pmatrix} s & -q \\ -r & p \end{pmatrix} \]
This means we interchange the diagonal elements and change the signs of the off-diagonal elements.

3. **Apply the Property**:
- Interchange the diagonal elements \( a \) and \( d \):
- The new diagonal elements will be \( d \) (in place of \( a \)) and \( a \) (in place of \( d \)).
- Change the signs of the off-diagonal elements \( b \) and \( c \):
- The new off-diagonal elements will be \( -b \) and \( -c \).

Therefore, the adjoint of \( S \) becomes:
\[ \text{adj}(S) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

4. **Final Result**:
Thus, the adjoint of the matrix \( S \) is:
\[ \text{adj}(S) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]