Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 46 / 1 / i / Solution

Put $D_1=X_1-Y_1$, $D_2=X_2-Y_2$ andThe difference vector is $D=BZ$. Its mean is zero because the two sons have the same mean vector. Its covariance matrix isFor example, $\mathrm{Cov}(D_1,D_2)=b-c-c+b=2(b-c)$. ThusThe positive definiteness assumption supplies $a-b>0$ and $a+b-2c>0$: these are eigenvalues of the original matrix on within-son contrast and between-son difference directions. In particular the resulting difference covariance is positive definite.