Right now I am working on my implementation of Knuth's DLX algorithm.

I did understand the exact cover problem. But I didn't fully understand how Dancing links works. Here is Knuth's paper. The question is about the solutions presented in the paper. I don't understand how these solutions make sense taking into account the matrix that he uses.

Matrix for the exact cover problem

     A B C D E F G
1  0 0 1 0 1 1 0
2  1 0 0 1 0 0 1
3  0 1 1 0 0 1 0
4  1 0 0 1 0 0 0
5  0 1 0 0 0 0 1
6  0 0 0 1 1 0 1


And solutions are:

A D
B G
C E F


But how come these solutions are valid? Isn't it supposed to be answered like A B E? Where in each row there is only one 1.