If `A=[{:(1,2,3),(2,3,2),(3,3,4):}]`, find `adj.A`

Answer

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

### Step 1: Find the Cofactors of the Matrix

The cofactor \( C_{ij} \) of an element \( a_{ij} \) is calculated using the formula:
\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]
where \( M_{ij} \) is the determinant of the submatrix obtained by deleting the \( i \)-th row and \( j \)-th column.

### Step 2: Calculate the Cofactors

1. **Cofactor \( C_{11} \)**:
\[ M_{11} = \begin{vmatrix} 3 & 2 \\ 3 & 4 \end{vmatrix} = (3)(4) - (2)(3) = 12 - 6 = 6 \]
\[ C_{11} = (-1)^{1+1} \cdot 6 = 6 \]

2. **Cofactor \( C_{12} \)**:
\[ M_{12} = \begin{vmatrix} 2 & 2 \\ 3 & 4 \end{vmatrix} = (2)(4) - (2)(3) = 8 - 6 = 2 \]
\[ C_{12} = (-1)^{1+2} \cdot 2 = -2 \]

3. **Cofactor \( C_{13} \)**:
\[ M_{13} = \begin{vmatrix} 2 & 3 \\ 3 & 3 \end{vmatrix} = (2)(3) - (3)(3) = 6 - 9 = -3 \]
\[ C_{13} = (-1)^{1+3} \cdot (-3) = -3 \]

4. **Cofactor \( C_{21} \)**:
\[ M_{21} = \begin{vmatrix} 2 & 3 \\ 3 & 4 \end{vmatrix} = (2)(4) - (3)(3) = 8 - 9 = -1 \]
\[ C_{21} = (-1)^{2+1} \cdot (-1) = 1 \]

5. **Cofactor \( C_{22} \)**:
\[ M_{22} = \begin{vmatrix} 1 & 3 \\ 3 & 4 \end{vmatrix} = (1)(4) - (3)(3) = 4 - 9 = -5 \]
\[ C_{22} = (-1)^{2+2} \cdot (-5) = -5 \]

6. **Cofactor \( C_{23} \)**:
\[ M_{23} = \begin{vmatrix} 1 & 2 \\ 3 & 3 \end{vmatrix} = (1)(3) - (2)(3) = 3 - 6 = -3 \]
\[ C_{23} = (-1)^{2+3} \cdot (-3) = 3 \]

7. **Cofactor \( C_{31} \)**:
\[ M_{31} = \begin{vmatrix} 2 & 3 \\ 3 & 2 \end{vmatrix} = (2)(2) - (3)(3) = 4 - 9 = -5 \]
\[ C_{31} = (-1)^{3+1} \cdot (-5) = -5 \]

8. **Cofactor \( C_{32} \)**:
\[ M_{32} = \begin{vmatrix} 1 & 3 \\ 2 & 2 \end{vmatrix} = (1)(2) - (3)(2) = 2 - 6 = -4 \]
\[ C_{32} = (-1)^{3+2} \cdot (-4) = 4 \]

9. **Cofactor \( C_{33} \)**:
\[ M_{33} = \begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix} = (1)(3) - (2)(2) = 3 - 4 = -1 \]
\[ C_{33} = (-1)^{3+3} \cdot (-1) = -1 \]

### Step 3: Form the Cofactor Matrix

The cofactor matrix \( C \) is:
\[ C = \begin{bmatrix} 6 & -2 & -3 \\ 1 & -5 & 3 \\ -5 & 4 & -1 \end{bmatrix} \]

### Step 4: Transpose the Cofactor Matrix

The adjoint of \( A \), denoted as \( \text{adj}(A) \), is the transpose of the cofactor matrix:
\[ \text{adj}(A) = C^T = \begin{bmatrix} 6 & 1 & -5 \\ -2 & -5 & 4 \\ -3 & 3 & -1 \end{bmatrix} \]

### Final Answer

Thus, the adjoint of the matrix \( A \) is:
\[ \text{adj}(A) = \begin{bmatrix} 6 & 1 & -5 \\ -2 & -5 & 4 \\ -3 & 3 & -1 \end{bmatrix} \]