24/6 DESIGN FOR A BRAIN 



combine by the rules 



R + R = R, 0+0=0, # + 0=0 + ]? = #, 

 R x R = R, 0x0=0, Rx0=0xR=0. 



A sum of such elements can therefore be zero in general only if 



each element is zero. 



24/6. In an 7?0-matrix of order n x n, the zeros are patterned 

 if, given any zero not in the principal diagonal, we can separate 

 the numbers 1, 2, . . . , n into two sets a and /? (neither being 

 void) so that the minor left after suppressing columns a and rows /? 

 is composed wholly of zeros which include the given zero. For 

 example, the 720-matrix 







R R 



R 







R 

 R 

 R 



R 



has its zeros patterned. Selecting, for instance, 

 the third row, we can make a = 1, 3, 4 and /? = 2. 

 the minor 







the zero in 

 This leaves 



. 

 . 



where dots indicate eliminated elements ; the remaining elements 

 are all zero, and they include the selected zero. The other zeros 

 in the original matrix can all be treated similarly. 



24/7. Some necessary theorems will now be stated. Their proofs 

 are simple and need not be given here. 

 A matrix A is idempotent if A 2 = A. 

 Theorem : If an 7?0-matrix has no zeros in the principal diagonal 

 a necessary and sufficient condition that the zeros be patterned 

 is that the matrix be idempotent. 



24/8. Theorem : If A is an i?0-matrix of order n X w, and I 

 is the matrix with 72's in the principal diagonal and zeros elsewhere, 

 then the matrix 



I + A + A 2 + . . . + A n ~ x 

 is idempotent. 



244 



