In the determinant `|{:(5,-1,4),(2,3,-5),(-1,-2,6):}|`, find the co-factos of the elements `-5`, `3` and `6`

Answer

To find the cofactors of the elements -5, 3, and 6 in the determinant

\[ D = \begin{vmatrix} 5 & -1 & 4 \\ 2 & 3 & -5 \\ -1 & -2 & 6 \end{vmatrix} \]

we will follow the steps outlined below:

### Step 1: Find the cofactor of -5

The element -5 is located in the 2nd row and 3rd column. The formula for the cofactor \( C_{ij} \) is given by:

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

where \( M_{ij} \) is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column.

For -5 (i=2, j=3):

1. Remove the 2nd row and 3rd column from the determinant:

\[ \begin{vmatrix} 5 & -1 \\ -1 & -2 \end{vmatrix} \]

2. Calculate the determinant:

\[ M_{23} = (5)(-2) - (-1)(-1) = -10 - 1 = -11 \]

3. Apply the sign factor:

\[ C_{23} = (-1)^{2+3} \cdot (-11) = -1 \cdot (-11) = 11 \]

### Step 2: Find the cofactor of 3

The element 3 is located in the 2nd row and 2nd column.

For 3 (i=2, j=2):

1. Remove the 2nd row and 2nd column from the determinant:

\[ \begin{vmatrix} 5 & 4 \\ -1 & 6 \end{vmatrix} \]

2. Calculate the determinant:

\[ M_{22} = (5)(6) - (4)(-1) = 30 + 4 = 34 \]

3. Apply the sign factor:

\[ C_{22} = (-1)^{2+2} \cdot 34 = 1 \cdot 34 = 34 \]

### Step 3: Find the cofactor of 6

The element 6 is located in the 3rd row and 3rd column.

For 6 (i=3, j=3):

1. Remove the 3rd row and 3rd column from the determinant:

\[ \begin{vmatrix} 5 & -1 \\ 2 & 3 \end{vmatrix} \]

2. Calculate the determinant:

\[ M_{33} = (5)(3) - (-1)(2) = 15 + 2 = 17 \]

3. Apply the sign factor:

\[ C_{33} = (-1)^{3+3} \cdot 17 = 1 \cdot 17 = 17 \]

### Final Results

- The cofactor of -5 is **11**.
- The cofactor of 3 is **34**.
- The cofactor of 6 is **17**.