Evaluate the following determinants by two method :
`|{:(0,2,0),(2,3,4),(4,5,6):}|`

Answer

To evaluate the determinant \( | \begin{pmatrix} 0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{pmatrix} | \) using two methods, we will follow these steps:

### Method 1: General Method

1. **Write the determinant**:
\[ D = \begin{vmatrix} 0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{vmatrix} \]

2. **Expand the determinant using the first row**:
\[ D = 0 \cdot \text{Cofactor}(1,1) + 2 \cdot \text{Cofactor}(1,2) + 0 \cdot \text{Cofactor}(1,3) \]
Since the first and third terms are multiplied by 0, we only need to calculate the second term:
\[ D = 2 \cdot \text{Cofactor}(1,2) \]

3. **Calculate the Cofactor(1,2)**:
The cofactor is calculated as:
\[ \text{Cofactor}(1,2) = (-1)^{1+2} \cdot \begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} \]
Calculate the 2x2 determinant:
\[ \begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} = (2 \cdot 6) - (4 \cdot 4) = 12 - 16 = -4 \]
Thus,
\[ \text{Cofactor}(1,2) = -(-4) = 4 \]

4. **Substitute back into the determinant**:
\[ D = 2 \cdot 4 = 8 \]

### Method 2: Shortcut Method

1. **Write the determinant**:
\[ D = \begin{vmatrix} 0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{vmatrix} \]

2. **Use the shortcut method**:
- Write down the elements of the determinant:
- \( a_1 = 0, a_2 = 2, a_3 = 0 \)
- \( b_1 = 2, b_2 = 3, b_3 = 4 \)
- \( c_1 = 4, c_2 = 5, c_3 = 6 \)

3. **Calculate using the formula**:
\[ D = a_1b_2c_3 + a_2b_3c_1 + a_3b_1c_2 - (a_3b_2c_1 + a_2b_1c_3 + a_1b_3c_2) \]
Substituting the values:
\[ D = 0 \cdot 3 \cdot 6 + 2 \cdot 4 \cdot 4 + 0 \cdot 2 \cdot 5 - (0 \cdot 3 \cdot 4 + 2 \cdot 2 \cdot 6 + 0 \cdot 4 \cdot 5) \]
Simplifying:
\[ D = 0 + 32 + 0 - (0 + 24 + 0) = 32 - 24 = 8 \]

### Final Result
Thus, the value of the determinant is:
\[ \boxed{8} \]