I am reading: Boolean Matrix Theory by K.H. Kim. On page 37 the definition for Schein rank is given.
Def 1.4.1: For vectors $v,w$ the symbol $c(v,w)$ will denote the matrix $(v_i w_j)$. Such matrices are called cross-vectors.
Def 1.4.2: The Schein rank of a matrix $A$ is the least number of cross-vectors whose sum is $A$.
Example 1.4.1 $A = \left( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ \end{array} \right)$
The text states the Schein rank of $A$ is $3$.
Why is the Schein rank equal to $3$? What $3 $ cross-vectors sum to $A$.