Find the values of the following determinants
`|{:(13,15,17),(14,16,18),(15,17,19):}|`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 13 & 15 & 17 \\ 14 & 16 & 18 \\ 15 & 17 & 19 \end{vmatrix} \]

we can use properties of determinants to simplify our calculations.

### Step 1: Apply Row Operations
We can use the property of determinants that allows us to add or subtract multiples of one row to another row without changing the value of the determinant.

Let's perform the operation \( R_1 \leftarrow R_1 + R_3 - 2R_2 \).

Calculating each element of the new \( R_1 \):

- First element: \( 13 + 15 - 2 \times 14 = 13 + 15 - 28 = 0 \)
- Second element: \( 15 + 17 - 2 \times 16 = 15 + 17 - 32 = 0 \)
- Third element: \( 17 + 19 - 2 \times 18 = 17 + 19 - 36 = 0 \)

Thus, the new matrix becomes:

\[ D = \begin{vmatrix} 0 & 0 & 0 \\ 14 & 16 & 18 \\ 15 & 17 & 19 \end{vmatrix} \]

### Step 2: Evaluate the Determinant
Now, we can see that the first row of the determinant is all zeros. A property of determinants states that if any row (or column) of a determinant is entirely zero, the value of the determinant is zero.

Thus, we conclude:

\[ D = 0 \]

### Final Answer
The value of the determinant is \( 0 \).

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