Use properties of determinants to evaluate :
`|{:(2,3,1),(4,6,2),(1,3,2):}|`

Answer

To evaluate the determinant \( | \begin{vmatrix} 2 & 3 & 1 \\ 4 & 6 & 2 \\ 1 & 3 & 2 \end{vmatrix} | \) using properties of determinants, we can follow these steps:

### Step 1: Factor out common elements
We can factor out the common element from the second row. The second row is \( (4, 6, 2) \), which can be factored as \( 2(2, 3, 1) \).

Thus, we can rewrite the determinant as:
\[ | \begin{vmatrix} 2 & 3 & 1 \\ 2(2 & 3 & 1) \\ 1 & 3 & 2 \end{vmatrix} | = 2 | \begin{vmatrix} 2 & 3 & 1 \\ 2 & 3 & 1 \\ 1 & 3 & 2 \end{vmatrix} | \]

### Step 2: Identify identical rows
Now we notice that the first row \( (2, 3, 1) \) and the second row \( (2, 3, 1) \) are identical.

### Step 3: Apply the property of determinants
According to the properties of determinants, if two rows (or columns) of a determinant are identical, the value of the determinant is zero. Therefore:
\[ | \begin{vmatrix} 2 & 3 & 1 \\ 2 & 3 & 1 \\ 1 & 3 & 2 \end{vmatrix} | = 0 \]

### Step 4: Conclude the evaluation
Thus, we have:
\[ 2 | \begin{vmatrix} 2 & 3 & 1 \\ 2 & 3 & 1 \\ 1 & 3 & 2 \end{vmatrix} | = 2 \cdot 0 = 0 \]

Therefore, the value of the determinant \( | \begin{vmatrix} 2 & 3 & 1 \\ 4 & 6 & 2 \\ 1 & 3 & 2 \end{vmatrix} | \) is:
\[ \boxed{0} \]