# Calculate Upper Lower bound in $4\times 4$ correlation matrix

Let $X_i$, $i = 1,2,3 4$, be random variables on the same probability space such that \begin{align*} \mathrm{corr}(X_1,X_3) &= 0.3;\\ \mathrm{corr}(X_2,X_3) &= 0.1;\\ \mathrm{corr}(X_1,X_4) &= 0.2;\\ \mathrm{corr}(X_2,X_4) &= −0.1;\\ \mathrm{corr}(X_3,X_4) &= −0.2. \end{align*} Find upper and lower bounds for $\mathrm{corr}(X_1,X_2)$.

Any help on how to approach this will be appreciated.

I have been able to create the partial matrix, but not sure how to proceed from here.

$$\mathrm{corr}(x_1,x_2)=\mathrm{corr}(x_2,x_1)=x$$ $$\left(\begin{array}{rrrr} 1 & x & 0.3 & 0.2\\ x & 1 & 0.1 & -0.1\\ 0.3& 0.1& 1 & -0.2\\ 0.2 &-0.1 &-0.2& 1 \end{array}\right)$$

A correlation matrix $C$ must be positive definite, which means that it must satisfy $\det(C)>0$. For your case you have
$$C = \left[ \begin{array}{rrrr} 1 & x & 0.3 & 0.2 \\ x & 1 & 0.1 & -0.1 \\ 0.3 & 0.1 & 1 & -0.2 \\ 0.2 & -0.1& -0.2 & 1 \end{array}\right]$$
$$\det(C) = 0.7925+0.016 x-0.96 x^2$$
You can now find the roots of this quadratic, which will tell you where the upper and lower bounds on $x$ are.