Evaluate `|[0, 2, 0], [2, 3, 4], [4, 5, 6]|`

Explanation

To evaluate the determinant of the matrix

\[ \begin{vmatrix} 0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{vmatrix} \]

we can follow these steps:

### Step 1: Set up the determinant
We have the matrix:

\[ A = \begin{bmatrix} 0 & 2 & 0 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{bmatrix} \]

### Step 2: Expand the determinant along the first row
Since the first row contains two zeros, it is convenient to expand along this row. The formula for the determinant when expanding along the first row is:

\[ |A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]

where \(C_{ij}\) is the cofactor of the element \(a_{ij}\).

### Step 3: Calculate the cofactors
- For \(a_{11} = 0\): The term is \(0 \cdot C_{11} = 0\).
- For \(a_{12} = 2\): We calculate the cofactor \(C_{12}\):

\[ C_{12} = (-1)^{1+2} \begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} = -\begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} \]

Calculating this 2x2 determinant:

\[ \begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} = (2 \cdot 6) - (4 \cdot 4) = 12 - 16 = -4 \]

Thus,

\[ C_{12} = -(-4) = 4 \]

So, the contribution from this term is:

\[ 2 \cdot C_{12} = 2 \cdot 4 = 8 \]

- For \(a_{13} = 0\): The term is \(0 \cdot C_{13} = 0\).

### Step 4: Combine the contributions
Now we can combine the contributions:

\[ |A| = 0 + 8 + 0 = 8 \]

### Final Result
Thus, the value of the determinant is

\[ \boxed{8} \]