Find all the co-factors of `|{:(1,3),(-2,4):}|`

Answer

To find all the co-factors of the determinant given by the matrix:

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

we will follow the steps to calculate the co-factors \( C_{ij} \) for each element in the matrix.

### Step 1: Identify the matrix and its elements
The matrix is:

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

The elements of the matrix are:
- \( a_{11} = 1 \)
- \( a_{12} = 3 \)
- \( a_{21} = -2 \)
- \( a_{22} = 4 \)

### Step 2: Calculate the co-factor \( C_{11} \)
The co-factor \( C_{11} \) is calculated using the formula:

\[ C_{ij} = (-1)^{i+j} \cdot M_{ij} \]

where \( M_{ij} \) is the minor of the element \( a_{ij} \).

For \( C_{11} \):
- \( i = 1, j = 1 \)
- The minor \( M_{11} \) is the determinant of the matrix obtained by removing the first row and first column:

\[ M_{11} = \begin{vmatrix} 4 \end{vmatrix} = 4 \]

Thus,

\[ C_{11} = (-1)^{1+1} \cdot 4 = 1 \cdot 4 = 4 \]

### Step 3: Calculate the co-factor \( C_{12} \)
For \( C_{12} \):
- \( i = 1, j = 2 \)
- The minor \( M_{12} \) is the determinant of the matrix obtained by removing the first row and second column:

\[ M_{12} = \begin{vmatrix} -2 \end{vmatrix} = -2 \]

Thus,

\[ C_{12} = (-1)^{1+2} \cdot (-2) = -1 \cdot (-2) = 2 \]

### Step 4: Calculate the co-factor \( C_{21} \)
For \( C_{21} \):
- \( i = 2, j = 1 \)
- The minor \( M_{21} \) is the determinant of the matrix obtained by removing the second row and first column:

\[ M_{21} = \begin{vmatrix} 3 \end{vmatrix} = 3 \]

Thus,

\[ C_{21} = (-1)^{2+1} \cdot 3 = -1 \cdot 3 = -3 \]

### Step 5: Calculate the co-factor \( C_{22} \)
For \( C_{22} \):
- \( i = 2, j = 2 \)
- The minor \( M_{22} \) is the determinant of the matrix obtained by removing the second row and second column:

\[ M_{22} = \begin{vmatrix} 1 \end{vmatrix} = 1 \]

Thus,

\[ C_{22} = (-1)^{2+2} \cdot 1 = 1 \cdot 1 = 1 \]

### Summary of the co-factors
The co-factors of the matrix are:
- \( C_{11} = 4 \)
- \( C_{12} = 2 \)
- \( C_{21} = -3 \)
- \( C_{22} = 1 \)

Thus, the co-factors of the matrix are:

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