For an (n times m) array of real numbers, show that the four quadratic forms, (m n

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For an \(n \times m\) array of real numbers, show that the four quadratic forms, \(m n \bar{Y}_{. .}^{2}\),

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are invariant with respect to row and column permutations. Here, \(Y_{i \text {. }}\) is the \(i\) th row total, and \(\bar{Y}_{i}\). is the row average.

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