3x3 Matrices with 2 numbers
| 0,0,0 (0) | 0,0,0 (1) | 0,0,0 (2) | 0,0,0 (3) | 0,0,0 (4) | 0,0,0 (5) | 0,0,0 (6) | 0,0,0 (7) | |
| a b c 0 0 0 0 0 0 |
a b c 0 0 0 0 0 1
|
a b c 0 0 0 0 1 0
|
a b c 0 0 0 0 1 1
|
a b c 0 0 0 1 0 0
|
a b c 0 0 0 1 0 1
|
a b c 0 0 0 1 1 0
|
a b c 0 0 0 1 1 1
|
|
| 0,0,0 (8) | 0,0,0 (9) | -1,0,0(10) | -1,0,0(11) | 0,-1,0(12) | 0,-1,0(13) | -1,-1,0(14) | -1,-1,0(15) | |
| a b c 0 0 1 0 0 0
|
a b c 0 0 1 0 0 1
|
a b c 0 0 1 0 1 0
|
a b c 0 0 1 0 1 1
|
a b c 0 0 1 1 0 0
|
a b c 0 0 1 1 0 1
|
a b c 0 0 1 1 1 0
|
a b c 0 0 1 1 1 1
|
|
| 0,0,0(16) | 1,0,0(17) | 0,0,0(18) | 1,0,0(19) | 0,0,-1(20) | 1,0,-1(21) | 0,0,-1(22) | 1,0,-1(23) | |
| a b c 0 1 0 0 0 0
|
a b c 0 1 0 0 0 1
|
a b c 0 1 0 0 1 0
|
a b c 0 1 0 0 1 1
|
a b c 0 1 0 1 0 0
|
a b c 0 1 0 1 0 1
|
a b c 0 1 0 1 1 0
|
a b c 0 1 0 1 1 1
|
|
| 0,0,0(24) | 1,0,0(25) | -1,0,0(26) | 0,0,0(27) | 0,-1,-1(28) | 1,-1,-1(29) | -1,-1,-1(30) | 0,-1,-1(31) | |
| a b c 0 1 1 0 0 0
|
a b c 0 1 1 0 0 1
|
a b c 0 1 1 0 1 0
|
a b c 0 1 1 0 1 1
|
a b c 0 1 1 1 0 0
|
a b c 0 1 1 1 0 1
|
a b c 0 1 1 1 1 0
|
a b c 0 1 1 1 1 1
|
|
| 0,0,0(32) | 0,1,0(33) | 0,0,1(34) | 0,1,1(35) | 0,0,0(36) | 0,1,0(37) | 0,0,1(38) | 0,1,1(39) | |
| a b c 1 0 0 0 0 0
|
a b c 1 0 0 0 0 1
|
a b c 1 0 0 0 1 0
|
a b c 1 0 0 0 1 1
|
a b c 1 0 0 1 0 0
|
a b c 1 0 0 1 0 1
|
a b c 1 0 0 1 1 0
|
a b c 1 0 0 1 1 1
|
|
| 0,0,0(40) | 0,1,0(41) | -1,0,1(42) | -1,1,1(43) | 0,-1,0(44) | 0,0,0(45) | -1,-1,1(46) | -1,0,1(47) | |
| a b c 1 0 1 0 0 0
|
a b c 1 0 1 0 0 1
|
a b c 1 0 1 0 1 0
|
a b c 1 0 1 0 1 1
|
a b c 1 0 1 1 0 0
|
a b c 1 0 1 1 0 1
|
a b c 1 0 1 1 1 0
|
a b c 1 0 1 1 1 1
|
|
| 0,0,0(48) | 1,1,0(49) | 0,0,1(50) | 1,1,1(51) | 0,0,-1(52) | 1,1,-1(53) | 0,0,0(54) | 1,1,0(55) | |
| a b c 1 1 0 0 0 0
|
a b c 1 1 0 0 0 1
|
a b c 1 1 0 0 1 0
|
a b c 1 1 0 0 1 1
|
a b c 1 1 0 1 0 0
|
a b c 1 1 0 1 0 1
|
a b c 1 1 0 1 1 0
|
a b c 1 1 0 1 1 1
|
|
| 0,0,0(56) | 1,1,0(57) | -1,0,1(58) | 0,1,1(59) | 0,-1,-1(60) | 1,0,-1(61) | -1,-1,0(62) | 0,0,0(63) | |
| a b c 1 1 1 0 0 0
|
a b c 1 1 1 0 0 1
|
a b c 1 1 1 0 1 0
|
a b c 1 1 1 0 1 1
|
a b c 1 1 1 1 0 0
|
a b c 1 1 1 1 0 1
|
a b c 1 1 1 1 1 0
|
a b c 1 1 1 1 1 1
|
|
| configuration | no. of matrices | configuration | no. of matrices | configuration | no. of matrices | |||
| 0,0,0 | 22 | 0,0,1 | 03 | 1,1,1 | 01 | |||
| 1,0,0 | 03 | 0,0,-1 | 03 | -1,-1,-1 | 01 | |||
| -1,0,0 | 03 | 0,1,1 | 03 | 1,1,-1 | 01 | |||
| 0,1,0 | 03 | 0,-1,-1 | 03 | -1,-1,1 | 01 | |||
| 0,-1,0 | 03 | 1,0,-1 | 03 | 1,-1,-1 | 01 | |||
| 1,1,0 | 03 | -1,0,1 | 03 | -1,1,1 | 01 | |||
| -1,-1,0 | 03 | sub-total | 58 | total | 64 (58+6) | |||
| *We have tried to construct a 3x3
matrix with 0,1 as its elements. Total no. of matrices is 29 =512.
The break up is as below:- no zero: 9C0=1 no 1: 9C0=1 1 zero:9C1=9 1 no. of 1: 9C1=9 2 zero:9C2=36 2 no. of 1: 9C2=36 3 zero: 9C3=84 3 no. of 1: 9C3=84 4 zero: 9C4=126 4 no. of 1: 9C4=126 ------------------256------------------------------256 5 zero: 9C5=126 5 no. of 1: 9C5=126 6 zero: 9C6=84 6 no. of 1: 9C6=84 7 zero: 9C7=36 7 no. of 1:9C7=36 8 zero:9C8=9 8 no. of 1:9C8=9 all 0 :9C9=1 all 1: 9C9=1 ------------------256--------------------------------256 TOTAL=512 Since no. of arrangement in a binary situation remains the same whether (for example) no. of zeroes is 2 or no. of ones is 2 , i.e 36, on interchanging 1 with a zero does not alter the no. of arrangement. That is if 1's and zeros are continually being exchanged, the arrangement remains stable. In other words, if arrangement remains the same, one cannot say with certainty whether it is 0 or 1. That means, the objects become indistinguishable like Bosonic particles. Handedness or chirality is already inbuilt in a matrix representation barring specific cases of symmetric matrices which commute. *The first row is variable a,b,c value being either 1 or 0. Barring first row, the no. of elements is 6 and therefore total possible configuration is 26 =64. We have drawn all 64 configuration above with 3 elements in 2nd row and 3 elements in 3rd row. The model is along the line of binary representation of codons, the numbers representing the decimal equivalent of binary starting from RHS of 3rd row proceeding towards left and likewise in 2nd row. * If the matrix is a b c a2 b2 c2 a3 b3 c3 determinant is a(b2c3-b3c2) - b(a2c3-a3c2) + c(a2b3-a3b2) = aA -bB +cC. In each matrix, A,B,C is computed. For example, 1,1,0 represents A=1,B=1 & C=1. * In some matrices, determinant is zero irrespective of a,b,c being zero or not. No. of such matrices is 22. * In other matrices, determinant value will depend on value ascribed to a,b,c. These values can be [-3,-2,-1,0,1,2,3] |
||||||||
| 0,0,0 (0) | -2,-2, 0 (1) | 2,0,-2 (2) | 0,-2,-2 (3) | 0,2,2 (4) | -2,0,2 (5) | 2,2,0 (6) | 0,0,0 (7) |
| a b c -1 -1 -1 -1 -1 -1 |
a b c -1 -1 -1 -1 -1 1
|
a b c -1 -1 -1 -1 1 -1
|
a b c -1 -1 -1 -1 1 1
|
a b c -1 -1 -1 1 -1 -1
|
a b c -1 -1 -1 1 -1 1
|
a b c -1 -1 -1 1 1 -1
|
a b c -1 -1 -1 1 1 1
|
| 2,2,0 (8) | 0,0,0 (9) | 0,2,-2(10) | -2,0,-2(11) | 2,0,2(12) | 0,-2,2(13) | 0,0,0(14) | -2,-2,0(15) |
| a b c -1 -1 1 -1 -1 -1
|
a b c -1 -1 1 -1 -1 1
|
a b c -1 -1 1 -1 1 -1
|
a b c -1 -1 1 -1 1 1
|
a b c -1 -1 1 1 -1 -1
|
a b c -1 -1 1 -1 -1 1
|
a b c -1 -1 1 1 1 -1
|
a b c -1 -1 1 1 1 1
|
| -2,0,2(16) | 0,-2,2(17) | 0,0,0(18) | 2,-2,0(19) | -2,2,0(20) | 0,0,0(21) | 0,2,-2(22) | 2,0,-2(23) |
| a b c -1 1 -1 -1 -1 -1
|
a b c -1 1 -1 -1 -1 1
|
a b c -1 1 -1 -1 1 -1
|
a b c -1 1 -1 -1 1 1
|
a b c -1 1 -1 1 -1 -1
|
a b c -1 1 -1 1 -1 1
|
a b c -1 1 -1 1 1 -1
|
a b c -1 1 -1 1 1 1
|
| 0,2,2(24) | 2,0,2(25) | -2,2,0(26) | 0,0,0(27) | 0,0,0(28) | 2,-2,0(29) | -2,0,-2(30) | 0,-2,-2(31) |
| a b c -1 1 1 -1 -1 -1
|
a b c -1 1 1 -1 -1 1
|
a b c -1 1 1 -1 1 -1
|
a b c 0 1 1 0 1 1
|
a b c -1 1 1 1 -1 -1
|
a b c -1 1 1 1 -1 1
|
a b c -1 1 1 1 1 -1
|
a b c -1 1 1 1 1 1
|
| 0,-2,-2(32) | -2,0,-2(33) | 2,-2,0(34) | 0,0,0(35) | 0,0,0(36) | -2,2,0(37) | 2,0,2(38) | 0,2,2(39) |
| a b c 1 -1-1 -1-1-1
|
a b c 1-1 -1 -1-1 1
|
a b c 1 -1-1 -11-1
|
a b c 1-1-1 -11 1
|
a b c 1-1-1 1-1-1
|
a b c 1-1-1 1-1 1
|
a b c 1-1 -1 1 1-1
|
a b c 1-1-1 1 1 1
|
| 2,0,-2(40) | 0,2,-2(41) | 0,0,0(42) | -2,2,0(43) | 2,-2,0(44) | 0,0,0(45) | 0,-2,2(46) | -2,0,2(47) |
| a b c 1 -1 1 -1-1-1
|
a b c 1 -1 1 -1 -1 1
|
a b c 1-1 1 -1 1-1
|
a b c 1-1 1 -11 1
|
a b c 1-1 1 1-1-1
|
a b c 1-1 1 1-1 1
|
a b c 1-1 1 1 1-1
|
a b c 1-1 1 1 1 1
|
| -2,-2,0(48) | 0,0,0(49) | 0,-2,2(50) | 2,0,2(51) | -2,0,-2(52) | 0,2,-2(53) | 0,0,0(54) | 2,2,0(55) |
| a b c 1 1 -1 -1-1-1
|
a b c 1 1 -1 -1 -1 1
|
a b c 1 1 -1 -1 1-1
|
a b c 1 1 -1 -1 1 1
|
a b c 1 1-1 1-1-1
|
a b c 1 1 -1 1-1 1
|
a b c 1 1-1 1 1-1
|
a b c 1 1 -1 1 1 1
|
| 0,0,0(56) | 2,2,0(57) | -2,0,2(58) | 0,2,2(59) | 0,-2,-2(60) | 2,0,-2(61) | -2,-2,0(62) | 0,0,0(63) |
| a b c 1 1 1 -1 -1 -1
|
a b c 1 1 1 -1 -1 1
|
a b c 1 1 1 -1 1 -1
|
a b c 1 1 1 -1 1 1
|
a b c 1 1 1 1 -1 -1
|
a b c 1 1 1 1-1 1
|
a b c 1 1 1 1 1 -1
|
a b c 1 1 1 1 1 1
|
| * Here these are 3x3 matrices where
each element can take value 1 or -1. Keeping the first row as a,b,c
, no. of elements is 6 and configuration is again 64. * We have drawn all 64 configuration in the line of first set of 3x3 matrices of 0,1 by substituting 0 with -1. * No. of configuration with A,B,C all zero is 16 where irrespective of value of a,b,c the determinant is zero. Then, no. of configuration where only A is zero is 16, Then, no. of configuration where only B is zero is 16, Then, no. of configuration where only C is zero is 16. * Except 0,0,0 configuration, everywhere , 2 out of A,B,C is non-zero and one is zero. A, B, C is either 0 or 2 or -2. Hence the determinant value is restricted to 0, 4 or -4 only. * the numbers 0-64 in brackets are redundant but kept to map with corresponding matrices in the previous chart. |
|||||||
| configuration | no. of matrices | configuration | no. of matrices | ||||
| 0,0,0 | 16 | 2,0,-2 | 04 | ||||
| 2,2,0 | 04 | -2,0,2 | 04 | ||||
| -2,-2,0 | 04 | 2,-2,0 | 04 | ||||
| 0,2,2 | 04 | -2,2,0 | 04 | ||||
| 0,-2,-2 | 04 | 0,-2,2 | 04 | ||||
| 2,0,2 | 04 | 0,2,-2 | 04 | ||||
| -2,0,-2 | 04 | total | 64 | ||||