Yesterday Only I learned about Fisher's inequality and I think it is good example to show application of rank calculations.

The problem is following:

Fisher, a population geneticist and statistician, was concerned with the design of experiments studying the differences among several different varieties of plants, under each of a number of different growing conditions, called "blocks".

Let:

 v be the number of varieties of plants;
b be the number of blocks.


It was required that:

1 k different varieties are in each block, k < v; no variety occurs twice in any one block;

2 any two varieties occur together in exactly λ blocks;

3 each variety occurs in exactly r blocks.


Fisher's inequality states simply that $v \leq b$.

And its proof (given below) involves basic linear algebra.

Let the incidence matrix $M$ be a $v×b$ matrix defined so that $M_{i,j}$ is 1 if element $i$ is in block $j$ and $0$ otherwise. Then $B=MM^T$ is a $v×v$ matrix such that $B_{i,i}=r$ and $B_{i,j}=λ$ for $i \neq j$. Since $r\neq \lambda$, $det(B) \neq 0$, so $rank(B) = v$; on the other hand, $rank(B) \leq rank(M) \leq b$, so $v \leq b$.