If $$A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}$$ and $$B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}$$ be such that AB = $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$, then:

We are given two matrices: matrix A and matrix B. Matrix A is a 2x3 matrix defined as $$A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}$$ and matrix B is a 3x1 column vector defined as $$B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}$$. The product AB is given to be $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$. We need to find the relationship between y and x from the options.

To compute the product AB, we multiply matrix A by matrix B. Since A is 2x3 and B is 3x1, the result will be a 2x1 matrix. Each element of the product is obtained by taking the dot product of each row of A with the column vector B.

For the first element (top entry) of AB, we take the first row of A, which is [1, 2, x], and dot it with B, which is [y, x, 1]. This gives: $$(1 \times y) + (2 \times x) + (x \times 1) = y + 2x + x = y + 3x$$.

For the second element (bottom entry) of AB, we take the second row of A, which is [3, -1, 2], and dot it with B: $$(3 \times y) + (-1 \times x) + (2 \times 1) = 3y - x + 2$$.

Therefore, the product AB is: $$\begin{bmatrix} y + 3x \\ 3y - x + 2 \end{bmatrix}$$.

We are told that AB equals $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$. So we set up the following equations:

Equation 1: $$y + 3x = 6$$

Equation 2: $$3y - x + 2 = 8$$

Simplify Equation 2 by subtracting 2 from both sides: $$3y - x = 6$$.

Now we have the system:

$$y + 3x = 6 \quad \text{(Equation 1)}$$

$$3y - x = 6 \quad \text{(Equation 2)}$$

We can solve this system using elimination. Multiply Equation 1 by 3 to align coefficients:

$$3(y + 3x) = 3 \times 6$$

$$3y + 9x = 18 \quad \text{(Equation 3)}$$

Subtract Equation 2 from Equation 3:

$$(3y + 9x) - (3y - x) = 18 - 6$$

$$3y + 9x - 3y + x = 12$$

$$10x = 12$$

Divide both sides by 10: $$x = \frac{12}{10} = \frac{6}{5}$$.

Substitute $$x = \frac{6}{5}$$ into Equation 1 to find y:

$$y + 3 \times \frac{6}{5} = 6$$

$$y + \frac{18}{5} = 6$$

Convert 6 to a fraction with denominator 5: $$6 = \frac{30}{5}$$

$$y = \frac{30}{5} - \frac{18}{5} = \frac{12}{5}$$

So we have $$x = \frac{6}{5}$$ and $$y = \frac{12}{5}$$.

Now, check the relationship: $$\frac{y}{x} = \frac{12/5}{6/5} = \frac{12}{5} \times \frac{5}{6} = \frac{12}{6} = 2$$, which means $$y = 2x$$.

Verify by substituting into the original equations:

Equation 1: $$y + 3x = \frac{12}{5} + 3 \times \frac{6}{5} = \frac{12}{5} + \frac{18}{5} = \frac{30}{5} = 6$$, correct.

Equation 2: $$3y - x + 2 = 3 \times \frac{12}{5} - \frac{6}{5} + 2 = \frac{36}{5} - \frac{6}{5} + \frac{10}{5} = \frac{40}{5} = 8$$, correct.

Comparing with the options:

A. $$y = 2x$$

B. $$y = -2x$$

C. $$y = x$$

D. $$y = -x$$

Hence, the correct answer is Option A.