What is the determinant of the matrix $$\left( {\matrix{ x & y & {y + z} \cr z & x & {z + x} \cr y & z & {x + y} \cr } } \right)$$ ?

If A, B and C are the angles of a triangle and $$\left| {\matrix{ 1 & 1 & 1 \cr {1 + \sin A} & {1 + \sin B} & {1 + \sin C} \cr {\sin A + {{\sin }^2}A} & {\sin B + {{\sin }^2}B} & {\sin C + {{\sin }^2}C} \cr } } \right| = 0$$, then which one of the following is correct?

If x + a + b + c = 0, then what is the value of $$\left| {\matrix{ {x + a} & b & c \cr a & {x + b} & c \cr a & b & {x + c} \cr } } \right|$$ ?

What are the values of x that satisfy the equation $$\left| {\matrix{ x & 0 & 2 \cr {2x} & 2 & 1 \cr 1 & 1 & 1 \cr } } \right| + \left| {\matrix{ {3x} & 0 & 2 \cr {{x^2}} & 2 & 1 \cr 0 & 1 & 1 \cr } } \right| = 0$$ ?