Find the adjoint of the following matrices :
`[{:(2,-1),(4,3):}]`

Answer

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

### Step 1: Find the Minors
The minors are calculated by removing the row and column of each element and finding the determinant of the remaining matrix.

1. **Minor \( M_{11} \)** (for element \( a_{11} = 2 \)):
\[ M_{11} = \text{det} \begin{pmatrix} 3 \end{pmatrix} = 3 \]

2. **Minor \( M_{12} \)** (for element \( a_{12} = -1 \)):
\[ M_{12} = \text{det} \begin{pmatrix} 4 \end{pmatrix} = 4 \]

3. **Minor \( M_{21} \)** (for element \( a_{21} = 4 \)):
\[ M_{21} = \text{det} \begin{pmatrix} -1 \end{pmatrix} = -1 \]

4. **Minor \( M_{22} \)** (for element \( a_{22} = 3 \)):
\[ M_{22} = \text{det} \begin{pmatrix} 2 \end{pmatrix} = 2 \]

### Step 2: Find the Cofactors
The cofactors are calculated using the formula:
\[ C_{ij} = (-1)^{i+j} M_{ij} \]

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = (-1)^{1+1} M_{11} = 1 \cdot 3 = 3 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = (-1)^{1+2} M_{12} = -1 \cdot 4 = -4 \]

3. **Cofactor \( C_{21} \)**:
\[ C_{21} = (-1)^{2+1} M_{21} = -1 \cdot (-1) = 1 \]

4. **Cofactor \( C_{22} \)**:
\[ C_{22} = (-1)^{2+2} M_{22} = 1 \cdot 2 = 2 \]

### Step 3: Form the Cofactor Matrix
The cofactor matrix is:
\[ C = \begin{pmatrix} 3 & -4 \\ 1 & 2 \end{pmatrix} \]

### Step 4: Transpose the Cofactor Matrix
The adjoint of matrix \( A \) is the transpose of the cofactor matrix:
\[ \text{adj}(A) = C^T = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix} \]

### Final Answer
Thus, the adjoint of the matrix \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix} \]