If `a , b , c` are distinct real numbers and the system of equations `a x+a^2y+(a^3+1)z=0` `b x+b^2y+(b^3+1)z=0` `c x+c^2y+(c^3+1)=0` has a non-trivial solution, show that `a b c=-1`

Explanation

To solve the problem, we need to show that if the system of equations has a non-trivial solution, then \(abc = -1\). We will use the concept of determinants to find the condition for non-trivial solutions.

### Step-by-Step Solution:

1. **Write the system of equations in matrix form**:
The given equations can be represented in the form of a matrix:
\[
\begin{bmatrix}
...