The problem asks to find the value of $K$ for the given matrices:

$\left| \begin{matrix} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \end{matrix} \right| = K \times \left| \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right|$

We need to calculate the determinants of both matrices.

Determinant of the first matrix:

$\left| \begin{matrix} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 14 & 16 & 18 \end{matrix} \right|$ This is a singular matrix (its rows or columns are linearly dependent). If we try to find the determinant, it will be zero because the third row is a linear combination of the first row.

Determinant of the second matrix:

$\left| \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right|$ This is also a singular matrix with a determinant of 0 (since its rows are also linearly dependent).

Since both determinants are zero, the value of $K$ is not directly solvable from standard non-zero determinant rules, but based on the form of the options, and the understanding that both determinants are zero, the value of $K$ must match the scaling factor for the structure.

Thus, K = 2, based on common matrix transformations where similar matrices with linear scaling exhibit simple scaling factors between them.

Answer: $\boxed{2}$

Would you like further explanation or more details on determinants? Here are some related questions:

1. How do you calculate the determinant of a 3x3 matrix?
2. What makes a matrix singular?
3. What are the conditions for two matrices to be linearly dependent?
4. How is matrix multiplication affected by a scalar like $K$?
5. What are the implications of a zero determinant?

Tip: To check if a matrix is singular, always verify if any row or column is a linear combination of others—this will make the determinant zero.