Find the minor and cor-factor of each element of the first column of the following determinants :
`|{:(0,2,6),(1,5,0),(3,7,1):}|`
Hence or otherwise evaluate them.

Answer

To solve the problem, we will find the minors and cofactors of each element in the first column of the given determinant. The determinant is:

\[ \begin{vmatrix} 0 & 2 & 6 \\ 1 & 5 & 0 \\ 3 & 7 & 1 \end{vmatrix} \]

### Step 1: Find the Minor of the First Element (0)

The first element is \(0\) located at position \(a_{11}\). To find its minor \(M_{11}\), we remove the first row and first column:

\[ M_{11} = \begin{vmatrix} 5 & 0 \\ 7 & 1 \end{vmatrix} \]

Calculating the determinant:

\[ M_{11} = (5 \cdot 1) - (0 \cdot 7) = 5 - 0 = 5 \]

### Step 2: Find the Minor of the Second Element (1)

The second element is \(1\) located at position \(a_{21}\). To find its minor \(M_{21}\), we remove the second row and first column:

\[ M_{21} = \begin{vmatrix} 2 & 6 \\ 7 & 1 \end{vmatrix} \]

Calculating the determinant:

\[ M_{21} = (2 \cdot 1) - (6 \cdot 7) = 2 - 42 = -40 \]

### Step 3: Find the Minor of the Third Element (3)

The third element is \(3\) located at position \(a_{31}\). To find its minor \(M_{31}\), we remove the third row and first column:

\[ M_{31} = \begin{vmatrix} 2 & 6 \\ 5 & 0 \end{vmatrix} \]

Calculating the determinant:

\[ M_{31} = (2 \cdot 0) - (6 \cdot 5) = 0 - 30 = -30 \]

### Step 4: Find the Cofactor of Each Element

The cofactor \(C_{ij}\) is given by:

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

#### Cofactor of \(0\) (C_{11})

\[ C_{11} = (-1)^{1+1} M_{11} = 1 \cdot 5 = 5 \]

#### Cofactor of \(1\) (C_{21})

\[ C_{21} = (-1)^{2+1} M_{21} = -1 \cdot (-40) = 40 \]

#### Cofactor of \(3\) (C_{31})

\[ C_{31} = (-1)^{3+1} M_{31} = 1 \cdot (-30) = -30 \]

### Summary of Results

- Minor of \(0\) is \(5\) and Cofactor is \(5\).
- Minor of \(1\) is \(-40\) and Cofactor is \(40\).
- Minor of \(3\) is \(-30\) and Cofactor is \(-30\).

### Step 5: Evaluate the Determinant

Now, we can evaluate the determinant using the cofactors of the first column:

\[ \text{Determinant} = a_{11}C_{11} + a_{21}C_{21} + a_{31}C_{31} \]

Substituting the values:

\[ \text{Determinant} = 0 \cdot 5 + 1 \cdot 40 + 3 \cdot (-30) = 0 + 40 - 90 = -50 \]

### Final Answer

The value of the determinant is \(-50\).