In a variance-covariance matrix, which statement correctly describes the diagonals and off-diagonals?

• A. Diagonal elements represent covariances; off-diagonal elements represent variances.
• B. The diagonal elements are the means of the variables.
• C. All off-diagonal elements are zero.
• D. Diagonal elements represent variances; off-diagonal elements represent covariances.

Answer: (D)

In a variance-covariance matrix, each diagonal entry is the variance of a variable, reflecting its spread. Each off-diagonal entry is the covariance between a pair of variables, showing how they vary together. The matrix is symmetric because Cov(X_i, X_j) = Cov(X_j, X_i). Diagonal values are nonnegative, while off-diagonal values can be positive, negative, or zero depending on the direction and strength of the relationship. A positive covariance means the variables tend to increase together, a negative covariance means one tends to increase while the other decreases, and a zero covariance indicates no linear relationship. This combination—diagonals as variances and off-diagonals as covariances—is what defines the covariance matrix. (If you were looking at a correlation matrix, diagonals would be 1 and off-diagonals would be correlations rather than covariances.)