The value of a for which the system of equations

$a^3x + (a+1)^3 y + (a + 2)^3 z = 0$

$ax + (a + 1) y + (a + 2) z = 0$

$x + y + z = 0$

has a non-zero solution, is

-1

The given system of equations will have a non-zero solution, if

$\begin{vmatrix}a^3& (a+1)^3& (a+2)^3\\a &(a+1)& (a + 2)\\1&1&1\end{vmatrix}=0$

$⇒\begin{vmatrix}a^3 -(a + 2)^3& (a+1)^3 - (a + 2)^3& (a+2)^3\\-2&-1&a+2\\0&0&1\end{vmatrix}=0$  [Applying $C_1→C_1-C_3$ and $C_2 →C_2-C_3$]

$⇒- a^3 +(a + 2)^3 + 2 (a + 1)^3-2 (a + 2)^3 = 0$

$⇒(a+2)^3-2 (a+1)^3 + a^3 = 0$

$⇒(6a^2 + 12a +8) −2 (3a^2 + 3a + 1)=0$

$⇒6a+ 6 = 0$

$⇒a = -1$