I am sure this is an easy question but not sure where to start:
Currencies A and B have correlation of 0.8 and currencies B and C have correlation of 0.9, what is the least correlation between A and C? The hint says write down the 3x3 correlation and denote x for the correlation bewteen A and C then only the unknown term in $\displaystyle \Sigma$ is $\displaystyle \Sigma_{13}=\Sigma_{31}$. Since $\displaystyle \Sigma$ is a correltation matrix it is symmetric and positive definite. So determine the interval $\displaystyle (x_0, x_1)$ so that $\displaystyle \Sigma$ has eigenvalues >0 provded $\displaystyle x\epsilon (x_0,x_1)$. So apparently the answer is $\displaystyle x_0$. Can anyone shed any light on this question, I'm going round in circles but started with this: is this even right?

$\displaystyle \left(\begin{array}{ccc}1&0.8&x\\0.8&1&0.9\\x&0.9& 1\end{array}\right)$