Let n be the number obtained on rolling a fair die. If the...

The coefficient matrix has determinant $\det(A) = -(n-1)(n-2)$.
The system has a unique solution when $\det(A) \neq 0$, which requires $n \neq 1$ and $n \neq 2$.
For a fair die, $n \in \{1,2,3,4,5,6\}$, so the unique solution occurs for $n \in \{3,4,5,6\}$.
The probability is $\frac{4}{6} = \frac{2}{3}$, giving $k = 4$.
The sum of $k$ and all possible values of $n$ where unique solution exists is $4 + (3+4+5+6) = 22$.