Matrix of co-factors of the matrix `[[1,2],[ 3, 4]]` is

Options

A. `[[4, -2], [-3, 1]]`
B. `[[4, -3], [-2, 1]]`
C. `[[-4, 2], [3, -1]]`
D. `[[-4, 3], [2, -1]]`

Answer

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

### Step 1: Identify the matrix and its elements
The given matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \]

### Step 2: Calculate the co-factors
The co-factor \( C_{ij} \) of an element \( a_{ij} \) in a matrix is calculated as:
\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]
where \( M_{ij} \) is the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column from \( A \).

#### a) Calculate \( C_{11} \)
- For \( C_{11} \), we remove the first row and first column:
\[ M_{11} = \text{det}\begin{bmatrix} 4 \end{bmatrix} = 4 \]
Thus,
\[ C_{11} = (-1)^{1+1} \cdot 4 = 4 \]

#### b) Calculate \( C_{12} \)
- For \( C_{12} \), we remove the first row and second column:
\[ M_{12} = \text{det}\begin{bmatrix} 3 \end{bmatrix} = 3 \]
Thus,
\[ C_{12} = (-1)^{1+2} \cdot 3 = -3 \]

#### c) Calculate \( C_{21} \)
- For \( C_{21} \), we remove the second row and first column:
\[ M_{21} = \text{det}\begin{bmatrix} 2 \end{bmatrix} = 2 \]
Thus,
\[ C_{21} = (-1)^{2+1} \cdot 2 = -2 \]

#### d) Calculate \( C_{22} \)
- For \( C_{22} \), we remove the second row and second column:
\[ M_{22} = \text{det}\begin{bmatrix} 1 \end{bmatrix} = 1 \]
Thus,
\[ C_{22} = (-1)^{2+2} \cdot 1 = 1 \]

### Step 3: Form the co-factor matrix
Now we can form the co-factor matrix \( C \):
\[ C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} \]

### Final Answer
The matrix of co-factors is:
\[ \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix} \]