Consider the matrix A=(111abca2b2c2)A = \begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{pmatrix}A=1aa21bb21cc2. For which values of a,b,ca, b, ca,b,c is det(A)=0\det(A) = 0det(A)=0?
If a=ba=ba=b or b=cb=cb=c or a=ca=ca=c
If a+b+c=0a+b+c = 0a+b+c=0
Only if a=b=c=0a=b=c=0a=b=c=0
If a,b,ca, b, ca,b,c are in arithmetic progression