Given system of equations has a non-trivial solution Π1≡x+ay+az=0 Π2≡bx+y+bz=0 Π3≡cx+cy+z=0 It can be written as AX=0 Where, ‌‌A=[
1
a
a
b
1
b
c
c
1
] So, the system has non-trivial solutions hence it has infinitely many solutions or unique solutions. ∴‌‌|A|=0 Now, let the system has infinitely many solutions So, let y=t;t∈R Then from Eqs. (i) and (ii), we get x+az=−at⇒bx+bz=−t Solving these, we get z=‌
t(1−ab)
b(a−1)
,x=‌
a
b
t‌
(1−b)
(1−a)
Hence, if a≠b≠c, then x=‌
a
b
t‌
(1−b)
(1−a)
,y=t, z=‌
t
b
‌
(1−ab)
(a−1)
,t∈R If a=b=c Let y=t1,z=t2,t1,t2∈R Then by Eq. (i) x=−a(t1+t2) So, the system of equations has unique solution.