A matrix `A=[(2,5),(-11,7)] , (abj A)' ` is equal to

Options

A. `[(-2,-5),(11,-7)]`
B. `[(7,5),(11,2)]`
C. `[(7,11),(-5,2)]`
D. `[(7,-5),(11,2)]`

Explanation

To find the adjoint of the matrix \( A = \begin{pmatrix} 2 & 5 \\ -11 & 7 \end{pmatrix} \) and then its transpose, we will follow these steps:

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

The cofactor \( C_{ij} \) is given by the formula:
\[ C_{ij} = (-1)^{i+j} M_{ij} \]
where \( M_{ij} \) is the minor of the element at position \( (i, j) \).

#### Finding Cofactor \( C_{11} \):
- Minor \( M_{11} \): Remove the first row and first column, leaving \( 7 \).
- Thus, \( C_{11} = (-1)^{1+1} M_{11} = 1 \cdot 7 = 7 \).

#### Finding Cofactor \( C_{12} \):
- Minor \( M_{12} \): Remove the first row and second column, leaving \( -11 \).
- Thus, \( C_{12} = (-1)^{1+2} M_{12} = -1 \cdot (-11) = 11 \).

#### Finding Cofactor \( C_{21} \):
- Minor \( M_{21} \): Remove the second row and first column, leaving \( 5 \).
- Thus, \( C_{21} = (-1)^{2+1} M_{21} = -1 \cdot 5 = -5 \).

#### Finding Cofactor \( C_{22} \):
- Minor \( M_{22} \): Remove the second row and second column, leaving \( 2 \).
- Thus, \( C_{22} = (-1)^{2+2} M_{22} = 1 \cdot 2 = 2 \).

### Step 2: Form the Cofactor Matrix
The cofactor matrix \( C \) is:
\[ C = \begin{pmatrix} 7 & 11 \\ -5 & 2 \end{pmatrix} \]

### Step 3: Find the Adjoint of Matrix A
The adjoint of matrix \( A \) is the transpose of the cofactor matrix:
\[ \text{Adj}(A) = C^T = \begin{pmatrix} 7 & -5 \\ 11 & 2 \end{pmatrix} \]

### Step 4: Find the Transpose of the Adjoint
The transpose of the adjoint matrix \( \text{Adj}(A) \) is:
\[ \text{Adj}(A)^T = \begin{pmatrix} 7 & 11 \\ -5 & 2 \end{pmatrix} \]

### Final Answer
Thus, the transpose of the adjoint of matrix \( A \) is:
\[ \begin{pmatrix} 7 & 11 \\ -5 & 2 \end{pmatrix} \]