Write the co-factor of `7` in `|{:(4,5,6),(5,6,7),(13,15,17):}|`

Answer

To find the co-factor of the element `7` in the determinant of the matrix

\[ \begin{vmatrix} 4 & 5 & 6 \\ 5 & 6 & 7 \\ 13 & 15 & 17 \end{vmatrix} \]

we will follow these steps:

### Step 1: Identify the position of the element `7`
The element `7` is located in the second row and third column of the matrix. Therefore, its position can be denoted as \( a_{23} \).

### Step 2: Calculate the minor of the element `7`
The minor of an element in a matrix is the determinant of the matrix that remains after removing the row and column of that element. For the element `7`, we will remove the second row and the third column.

The remaining matrix after removing the second row and third column is:

\[ \begin{vmatrix} 4 & 5 \\ 13 & 15 \end{vmatrix} \]

Now, we will calculate the determinant of this 2x2 matrix.

### Step 3: Calculate the determinant of the 2x2 matrix
The determinant of a 2x2 matrix

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

is calculated using the formula \( ad - bc \).

For our matrix:

\[ \begin{vmatrix} 4 & 5 \\ 13 & 15 \end{vmatrix} = (4)(15) - (5)(13) = 60 - 65 = -5 \]

### Step 4: Calculate the co-factor of the element `7`
The co-factor \( C_{ij} \) of an element \( a_{ij} \) is given by:

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

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

For the element `7`, we have \( i = 2 \) and \( j = 3 \). Therefore:

\[ C_{23} = (-1)^{2+3} \cdot M_{23} = (-1)^{5} \cdot (-5) = -1 \cdot (-5) = 5 \]

### Final Answer
The co-factor of the element `7` in the given determinant is \( 5 \).

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