The problem asks to calculate the values of $x$ and $y$, given a matrix expression:

$\left( \frac{x \ 1}{1 \ x - x} \right) + \left( \frac{x \ 2}{2 \ 1} \right) = \left( \frac{4 \ 3}{1 \ 3} \right)$

The equation involves the addition of two 2x2 matrices on the left-hand side, equating to another 2x2 matrix on the right-hand side.

Let’s break it down:

1. Left-hand side matrix:

   $\left( \frac{x \ 1}{1 \ x - x} \right) + \left( \frac{x \ 2}{2 \ 1} \right)$ This results in: $\left( \frac{2x \ 3}{3 \ 1} \right)$

2. Equating to the right-hand side matrix:

   $\left( \frac{4 \ 3}{1 \ 3} \right)$

Now, compare each corresponding element from the two matrices:

• For the first element: $2x = 4$, so $x = 2$
• For the second element: $3 = 3$ (which holds true)
• For the third element: $3 = 1$ (which holds true)
• For the fourth element: $1 = 3$