Explanation:

- Condition 1: a+b+c=0

- When the sum of the three variables is zero, their symmetric determinant vanishes.

- The determinant results in a homogeneous system.

- Condition 2: a³ + b³ + c³ = 3abc

- This condition is a known identity for determinant vanishing when a, b, and c follow this equation.

- It implies symmetry in the elements leading to a vanishing determinant.

- Condition 3: a² + b² + c² - ab - bc - ca = 0

- This condition indicates that a, b, and c satisfy a specific symmetric relation.

- The determinant is zero under this condition due to the vector dependencies caused by symmetry.

- Correct Option: 1, 2, and 3

- All of these conditions independently lead to the determinant vanishing.