If `a=[[-2, 6], [-5, 7]]`, then adjA=

Options

A. `[[7, -6], [5, -2]]`
B. `[[7, -5], [6, -2]]`
C. `[[2, -6], [5, -7]]`
D. `[[-2, 6], [-5, 7]]`

Answer

To find the adjoint of the matrix \( A = \begin{bmatrix} -2 & 6 \\ -5 & 7 \end{bmatrix} \), we will follow these steps:

### Step 1: Find the Cofactor Matrix
The cofactor matrix is derived from the original matrix by calculating the cofactors of each element. The cofactor \( C_{ij} \) of an element \( a_{ij} \) is given by:
\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]
where \( M_{ij} \) is the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column.

#### Calculation of Cofactors:
1. **Cofactor of \( -2 \) (element at (1,1))**:
- Minor \( M_{11} = 7 \) (determinant of the submatrix formed by removing the first row and first column)
- Cofactor \( C_{11} = (+1) \cdot 7 = 7 \)

2. **Cofactor of \( 6 \) (element at (1,2))**:
- Minor \( M_{12} = -5 \) (determinant of the submatrix formed by removing the first row and second column)
- Cofactor \( C_{12} = (-1) \cdot (-5) = 5 \)

3. **Cofactor of \( -5 \) (element at (2,1))**:
- Minor \( M_{21} = 6 \) (determinant of the submatrix formed by removing the second row and first column)
- Cofactor \( C_{21} = (-1) \cdot 6 = -6 \)

4. **Cofactor of \( 7 \) (element at (2,2))**:
- Minor \( M_{22} = -2 \) (determinant of the submatrix formed by removing the second row and second column)
- Cofactor \( C_{22} = (+1) \cdot (-2) = -2 \)

Thus, the cofactor matrix is:
\[ \text{Cofactor Matrix} = \begin{bmatrix} 7 & 5 \\ -6 & -2 \end{bmatrix} \]

### Step 2: Find the Adjoint of A
The adjoint of a matrix is the transpose of the cofactor matrix. Therefore, we will transpose the cofactor matrix obtained in Step 1.

\[ \text{Adjoint of A} = \text{Transpose of Cofactor Matrix} = \begin{bmatrix} 7 & 5 \\ -6 & -2 \end{bmatrix}^T = \begin{bmatrix} 7 & -6 \\ 5 & -2 \end{bmatrix} \]

### Final Answer
\[ \text{adj} A = \begin{bmatrix} 7 & -6 \\ 5 & -2 \end{bmatrix} \]

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