Discussion
Which of the following properties is not true about the incidence matrix M of a BIBD \((r, b, s, k, \lambda)\) ?
Answer: C
The property that does not hold is M is a symmetric matrix.
The paper lists the parameters as \((r, b, s, k, \lambda)\) in the position of the usual \((v, b, r, k, \lambda)\), so \(s\) plays the role of the replication number. The incidence matrix M has a row for each treatment and a column for each block, with entry 1 when the treatment occurs in the block. Then:
- each column (block) has exactly \(k\) ones, the block size;
- each row (treatment) has exactly \(s\) ones, the number of replications;
- two distinct rows share 1s in exactly \(\lambda\) columns, since every pair of treatments occurs together in \(\lambda\) blocks, which is the defining balance property.
M need not be square (there are usually more blocks than treatments), and even for a symmetric BIBD, where it is square, the matrix itself is generally not symmetric. So symmetry is the property that fails.
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