Rank

The rank of a Matrix of size $n \times n$ is an integer between $0$ and $n$ that represents the dimension of the space generated by the matrix.

A matrix containing only zeroes will have a zero rank as it generates the space containing only one point $\{0\}$

An $n$ by $n$ matrix that is inversible has rank $n$ as it can generate the whole space.

When multiplying two matrices, the rank of the result is the rank of the smallest one.

The function that associate a matrix to its rank is thus a morphism of monoids from $(M, \times)$ (the set of all matrices of a given size) to $(\mathbb{N},\min)$.