What Matrix is This? References for Structured Matrix

[patiobarbecue]

Homework Statement

Are there references of the following structured matrix?

A =
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
;

Say we have group A=1,2; group B=1,2, 3; the rows of above matrix shows all possible selections: one element from each group. Hence A corresponding to the list:
11
12
13
21
22
23
;


B =
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 1 0 0
1 0 0 1 0 0 1 0
1 0 0 1 0 0 0 1
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 1 0 0
0 1 1 0 0 0 1 0
0 1 1 0 0 0 0 1
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 1 0 0
0 1 0 0 1 0 1 0
0 1 0 0 1 0 0 1
;

group A=1,2; B=1,2,3; C=1,2,3; and the above matrix indicating the list:
111
112
113
121
122
123
131
132
133
211
212
213
221
222
223
231
232
233
;

Homework Equations

Say the number of elements in a group and number of groups are arbitrary, then the linear system Ax=b, b not zero sometimes have solution; most time it is over determined, and a least square solution is x = (A'A)^-1A'b if (A'A) is invertible, when it is invertible?

The Attempt at a Solution

(A'A) is invertible and its inverse is visually appealing:

0.25000 0.50000 -0.25000 0.00000
-0.25000 0.00000 0.25000 0.50000
0.50000 -0.25000 0.50000 -0.25000
0.00000 0.25000 0.00000 0.25000

(B'B) is singular from Octave.

It seems that is a pattern, but it is pretty hard to derive it. Is this matrix a well known matrix?

 

[patiobarbecue]

By rearrange rows of B, I got P, and M=P'*P seems to have a nice pattern, M is singular though. Is it obvious why it is singular?

# Created by Octave 3.2.3, Sat Jul 17 18:31:46 2010 EDT
# name: M
# type: matrix
# rows: 8
# columns: 8
9 0 3 3 3 3 3 3
0 9 3 3 3 3 3 3
3 3 6 0 0 2 2 2
3 3 0 6 0 2 2 2
3 3 0 0 6 2 2 2
3 3 2 2 2 6 0 0
3 3 2 2 2 0 6 0
3 3 2 2 2 0 0 6
# name: A
# type: matrix
# rows: 18
# columns: 8
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 1 0 0
1 0 0 1 0 0 1 0
1 0 0 1 0 0 0 1
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 1 0 0
0 1 1 0 0 0 1 0
0 1 1 0 0 0 0 1
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 1 0 0
0 1 0 0 1 0 1 0
0 1 0 0 1 0 0 1
# name: P
# type: matrix
# rows: 18
# columns: 8
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 0 0 1
1 0 0 1 0 0 1 0
1 0 0 1 0 1 0 0
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 0 0 1
0 1 1 0 0 0 1 0
0 1 1 0 0 1 0 0
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 0 0 1
0 1 0 0 1 0 1 0
0 1 0 0 1 1 0 0