If `A = {:[(2,-3),(4,1)]:}`, then adjoint of matrix `A` is

Options

A. `{:[(1,3),(-4,2)]:}`
B. `{:[(1,-3),(-4,2)]:}`
C. `{:[(1,3),(4,-2)]:}`
D. `{:[(-1,-3),(-4,2)]:}`

Answer

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

### Step 1: Find the Minors of Each Element
The minor of an element in a matrix is the determinant of the submatrix formed by deleting the row and column of that element.

1. **Minor of \( a_{11} = 2 \)**:
- Remove the first row and first column, leaving us with the element \( 1 \).
- Minor \( M_{11} = 1 \).

2. **Minor of \( a_{12} = -3 \)**:
- Remove the first row and second column, leaving us with the element \( 4 \).
- Minor \( M_{12} = 4 \).

3. **Minor of \( a_{21} = 4 \)**:
- Remove the second row and first column, leaving us with the element \( -3 \).
- Minor \( M_{21} = -3 \).

4. **Minor of \( a_{22} = 1 \)**:
- Remove the second row and second column, leaving us with the element \( 2 \).
- Minor \( M_{22} = 2 \).

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

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

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

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

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:
\[ C = \begin{pmatrix} 1 & -4 \\ 3 & 2 \end{pmatrix} \]

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

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

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