I need to solved this linear algebra equation with an unambiguous solution

Question

You know that $$\left[\begin{array}{rrr} a & b & b \\ b & c & -b\\ c & d & a \end{array}\right] *\left[\begin{array}{rrr} a \\ 1\\ b \end{array}\right]= \left[\begin{array}{rrr} 0 \\ 0\\ 0 \end{array}\right] $$ For which a, b, c, d have the system of equations $$\left[\begin{array}{rrr} a & b & c \\ b & c & d\\ b & -b & a \end{array}\right] *\left[\begin{array}{rrr} x \\ y\\ z \end{array}\right]= \left[\begin{array}{rrr} 1 \\ 0\\ d \end{array}\right] $$ unambiguous solution?

Answer 1

Let $A$ be the first matrix. The second one is $A^T$.

Matrix $A$ has visibly a nonzero element in its kernel (whatever $a$ and $b$), therefore isn't invertible; as a consequence $\det(A)=0$ which gives $\det(A^T)=0$.

The second system of equations having a determinant equal to $0$ has never a unique solution (either 0 solutions or an infinity of solutions).