When sample size = 3, why this matrix is non-singular?

Question

When sample size $n=3$ is the sufficient and necessary condition for the following matrix $T$ to be non-singular?

To be clear, assume $X_1, X_2, X_3\overset{i.i.d}{\sim}(0,\Sigma)$, where $\Sigma$ is a 3 by 3 positive definite symmatrix covariance matrix. $S$ is sample covariance matrix assuming mean is zero, i.e. $S = (X_1X_1^T+X_2X_2^T+X_3X_3^T)/3=\begin{pmatrix} s_{11} & s_{12}& s_{13}\\ s_{12} & s_{22}& s_{23}\\ s_{13} & s_{23}& s_{33} \end{pmatrix}$.

Why the following matrix is non-singular when $n=3$ and singular when $n<3$? $$T=\begin{pmatrix} s_{11} & s_{12}& s_{13}& 0 & 0 & 0\\ s_{12} & s_{22}+s_{11}&s_{23}&s_{12}&s_{13}&0\\ s_{13} & s_{23}&s_{33}+s_{11}&0&s_{12}&s_{13}\\ 0 & s_{12}&0&s_{22}&s_{23}&0\\ 0&s_{13} &s_{12}&s_{23}&s_{22}+s_{33}&s_{23}\\ 0&0&s_{13}&0&s_{23}&s_{33} \end{pmatrix} $$

Apparently, $T$ is made of by "stacking" $S$, for example, the principal submatrix of column 2,4,5 is $S$ (if removing diagonal terms $s_{11}$ and $s_{33}$ which are stacked from another principal submatrix of $S$).

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I am trying to prove by contradiction. If it is singular, I was hoping to show that the columns of $S$ are linearly dependent. But I got stuck. I hope this can be approved in a general way.

Answer 1

I solved it, seemingly.

By noticing $T$ is a Hessian of $$\sum_{i=1}^p\omega_{.i}^TS\omega_{i.},$$ where $\omega_{.i}$ is the $i^{th}$ row of $\Omega = \Sigma^{-1}$.

This is a sum of strictly convex quadratic forms (given $n=3$). Hence its Hessian is positive definite.

Answer 2

For the original question with $n=2$: by the Rule of Sarrus the $3\times 3$ matrix $T$ has determinant $$ \det(T)=\det(S)^2(s_{11}+s_{22}). $$ This is equal to zero if $\det(S)=0$, or $s_{11}+s_{22}=0$.

For $n=3$, $\det(S)$ is again a factor, and we have $$ \det(T)=\det(S)(s_{11}^2s_{22} + s_{11}^2s_{33} - s_{11}s_{12}^2 - s_{11}s_{13}^2 + s_{11}s_{22}^2 + 2s_{11}s_{22}s_{33} + s_{11}s_{33}^2 - s_{12}^2s_{22} - 2s_{12}s_{13}s_{23} - s_{13}^2s_{33} + s_{22}^2s_{33} - s_{22}s_{23}^2 + s_{22}s_{33}^2 - s_{23}^2s_{33}). $$