Discussion
A block design with \(v\) treatments and \(b\) blocks is connected. Hence the rank of \(c\) matrix is :
Answer: A
The rank of the C matrix is \(v - 1\).
The C matrix of a block design is \(C = R - NK^{-1}N'\). Its rows add to zero, so its rank is at most \(v - 1\). A design is connected exactly when every elementary treatment contrast is estimable, which happens if and only if \(\mathrm{rank}(C) = v - 1\).
Rank \(v\) is impossible because of the zero row sums, and \(b - 1\) or \(b\) refer to the number of blocks, which does not determine the rank of the treatment information matrix.
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