The adjoint of the matrix `A= [{:( 1,2),( 3,-5) :}]` is

Options

A. `[{:( -5,-2),( -3,1):}]`
B. ` [{:( 5,2),( 3,-1) :}]`
C. ` [{:( 5,-2),( 3,-1) :}]`
D. ` [{:( 5,-2),( -3,-1):}]`

Answer

To find the adjoint of the matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & -5 \end{pmatrix} \), we will follow the steps below:

### Step 1: Find the Cofactors
The adjoint of a matrix is the transpose of the cofactor matrix. We need to find the cofactors \( C_{ij} \) for each element of the matrix.

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = (-1)^{1+1} \cdot M_{11} = 1 \cdot M_{11} \]
where \( M_{11} \) is the minor of the element at position (1,1), which is obtained by deleting the first row and first column. Thus,
\[ M_{11} = -5 \quad \Rightarrow \quad C_{11} = 1 \cdot (-5) = -5 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = (-1)^{1+2} \cdot M_{12} = -1 \cdot M_{12} \]
where \( M_{12} \) is the minor of the element at position (1,2), which is obtained by deleting the first row and second column. Thus,
\[ M_{12} = 3 \quad \Rightarrow \quad C_{12} = -1 \cdot 3 = -3 \]

3. **Cofactor \( C_{21} \)**:
\[ C_{21} = (-1)^{2+1} \cdot M_{21} = -1 \cdot M_{21} \]
where \( M_{21} \) is the minor of the element at position (2,1), which is obtained by deleting the second row and first column. Thus,
\[ M_{21} = 2 \quad \Rightarrow \quad C_{21} = -1 \cdot 2 = -2 \]

4. **Cofactor \( C_{22} \)**:
\[ C_{22} = (-1)^{2+2} \cdot M_{22} = 1 \cdot M_{22} \]
where \( M_{22} \) is the minor of the element at position (2,2), which is obtained by deleting the second row and second column. Thus,
\[ M_{22} = 1 \quad \Rightarrow \quad C_{22} = 1 \cdot 1 = 1 \]

### Step 2: Form the Cofactor Matrix
Now we can form the cofactor matrix \( C \):
\[ C = \begin{pmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{pmatrix} = \begin{pmatrix} -5 & -3 \\ -2 & 1 \end{pmatrix} \]

### Step 3: Transpose the Cofactor Matrix
The adjoint of matrix \( A \) is the transpose of the cofactor matrix \( C \):
\[ \text{adj}(A) = C^T = \begin{pmatrix} -5 & -2 \\ -3 & 1 \end{pmatrix} \]

### Final Answer
Thus, the adjoint of the matrix \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} -5 & -2 \\ -3 & 1 \end{pmatrix} \]

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