If `A=[{:(,1,2),(,2,1):}]` then adj A=

Options

A. `[{:(,1,-2),(,-2,1):}]`
B. `[{:(,2,1),(,1,1):}]`
C. `[{:(,1,-2),(,-2,-1):}]`
D. `[{:(,-1,2),(,2,-1):}]`

Answer

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

### Step 1: Find the Cofactor Matrix

The cofactor matrix is obtained by calculating the cofactors of each element of the matrix \( A \).

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = (-1)^{1+1} \cdot M_{11} \]
where \( M_{11} \) is the minor of the element in the first row and first column. The minor is the determinant of the matrix obtained by deleting the first row and first column:
\[ M_{11} = \begin{vmatrix} 1 \end{vmatrix} = 1 \]
Therefore,
\[ C_{11} = 1 \cdot 1 = 1 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = (-1)^{1+2} \cdot M_{12} \]
where \( M_{12} \) is the minor of the element in the first row and second column:
\[ M_{12} = \begin{vmatrix} 2 \end{vmatrix} = 2 \]
Therefore,
\[ C_{12} = -1 \cdot 2 = -2 \]

3. **Cofactor \( C_{21} \)**:
\[ C_{21} = (-1)^{2+1} \cdot M_{21} \]
where \( M_{21} \) is the minor of the element in the second row and first column:
\[ M_{21} = \begin{vmatrix} 2 \end{vmatrix} = 2 \]
Therefore,
\[ C_{21} = -1 \cdot 2 = -2 \]

4. **Cofactor \( C_{22} \)**:
\[ C_{22} = (-1)^{2+2} \cdot M_{22} \]
where \( M_{22} \) is the minor of the element in the second row and second column:
\[ M_{22} = \begin{vmatrix} 1 \end{vmatrix} = 1 \]
Therefore,
\[ C_{22} = 1 \cdot 1 = 1 \]

Now, we can write the cofactor matrix:
\[ \text{Cofactor Matrix} = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} \]

### Step 2: Find the Adjoint Matrix

The adjoint of a matrix is the transpose of its cofactor matrix. Thus, we take the transpose of the cofactor matrix:
\[ \text{Adjoint} \, A = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix}^T = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} \]

### Final Answer
The adjoint of matrix \( A \) is:
\[ \text{adj} \, A = \begin{pmatrix} 1 & -2 \\ -2 & 1 \end{pmatrix} \]

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