In the variance-covariance matrix given below (rows and columns ordered \(P, Q, R, S\)), which one of the following pairs of elements has the highest correlation? \[ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 3.73 & 1.85 & 3.23 \\ 0 & 1.85 & 0.93 & 1.61 \\ 0 & 3.23 & 1.61 & 2.79 \end{bmatrix} \]
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The variable \(P\) has zero variance so any pair involving \(P\) is undefined; compute \(\rho_{ij} = Cov(i,j)/\sqrt{Var(i)Var(j)}\) for the remaining pairs and compare.
Step 1: Build the full correlation matrix from the covariance matrix. The correlation matrix is obtained as $R = D^{-1/2}\,\Sigma\,D^{-1/2}$, where $D$ is the diagonal matrix of variances and $\Sigma$ is the given covariance matrix; each entry becomes $\rho_{ij} = \Sigma_{ij}/\sqrt{\Sigma_{ii}\Sigma_{jj}}$. Step 2: Drop the $P$ row and column. Since $\Sigma_{PP} = 0$, $D^{-1/2}$ is undefined for $P$, so every pair involving $P$, including option (D) $P$-$R$, is excluded from consideration. Step 3: Normalize the remaining $3\times3$ sub-block for $Q, R, S$. Using $\sqrt{\Sigma_{QQ}} = \sqrt{3.73} \approx 1.931$, $\sqrt{\Sigma_{RR}} = \sqrt{0.93} \approx 0.964$, $\sqrt{\Sigma_{SS}} = \sqrt{2.79} \approx 1.670$, the normalized correlation sub-matrix works out to \[ \rho_{QR} = \frac{1.85}{1.931 \times 0.964} \approx 0.99,\quad \rho_{QS} = \frac{3.23}{1.931 \times 1.670} \approx 1.00,\quad \rho_{RS} = \frac{1.61}{0.964 \times 1.670} \approx 1.00 \] Step 4: Scan the off-diagonal entries of this normalized sub-matrix for the maximum. All entries are close to the upper bound of 1 for a correlation coefficient, with $\rho_{QS}$ working out marginally the largest among the three, so $Q$-$S$ is picked as the pair of highest correlation. \[ \boxed{\text{Option (A), } Q\text{-}S} \] Note: IIT Guwahati officially declared this question Marks-To-All (MTA) as it was found defective, since the $Q$-$S$ and $R$-$S$ correlations both come out essentially equal to 1.00 from the rounded data given, so marks were awarded to every candidate for this question.
