Can 3 random variables have pairwise correlation -1?

Question

Can 3 random variables X, Y, Z have pairwise correlation -1?

This seems to be easy question, but I just got confused over this.

Answer 1

No: if $X,Y,Z$ are your variables, then as $corr(X,Y)=-1$, first $Var \ X\neq 0$ and then

$$ Var\left(\frac X{\sqrt{Var \ X}}+ \frac Y{\sqrt{Var\ Y}}\right) = 2 + 2corr(X,Y)=0 $$ Then $$ \frac X{\sqrt{Var \ X}} = - \frac Y{\sqrt{Var\ Y}} = + \frac Z{\sqrt{Var\ Z}} = - \frac X{\sqrt{Var \ X}} $$by symetry, then $X=0$. But it is impossible because $Var \ X>0$.

If you replace correlation by covariance, then it is possible: for example, any gaussian vector with variance matrix $\bigl(\begin{smallmatrix} 8 &-1&-1\\ -1 & 8&-1\\ -1&-1&8 \end{smallmatrix} \bigr)$ (because such a matrix has $>0$ eigenvalues).