Let \(Y\) be an \(n \times m\) array of real-valued random variables with zero mean and covariance matrix

\[
\operatorname{cov}\left(Y_{i r}, Y_{j s}ight)=\sigma_{0}^{2} \delta_{i j} \delta_{r s}+\sigma_{1}^{2} \delta_{i j}+\sigma_{2}^{2} \delta_{r s}+\sigma_{3}^{2}
\]

for some non-negative coefficients \(\sigma_{0}^{2}, \ldots, \sigma_{3}^{2}\). Show that the covariance matrix is invariant with respect to the product group consisting of \(n\) ! permutations applied to rows and \(m\) ! permutations applied to columns. In other words, show that the \(n \times m\) matrix whose \((i, s)\)-component is \(Y_{\sigma(i), \tau(s)}\), has the same covariance matrix as \(Y\).