Discussion
Which of the following matrices is a covariance matrix ?
Answer: C
The covariance matrix is \(\begin{bmatrix} 2 & -2 \\ -2 & 3 \end{bmatrix}\).
A covariance matrix must be symmetric and positive semi-definite. For a 2 by 2 symmetric matrix this means non-negative diagonal entries and a non-negative determinant. For option C, the diagonal is 2 and 3 and
\[ \det = 2 \cdot 3 - (-2)^{2} = 2 \gt 0 \]
Option A has determinant \(0 - 1/4 \lt 0\). Option B is not symmetric. Option D has determinant \(4 - 9 \lt 0\), which would mean a correlation of -3/2, outside [-1, 1].
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