finding the inverse of a matrx

Question

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix.

From this link http://en.wikipedia.org/wiki/Hill_cipher,

| 6  24  1  |-I      |  8 5 10  |
| 13 16  10 |     =  |  21 8 21 |
| 20 17  15 |        |  21 12 8 |

but from my java program , the output of the inverse after using Gauss Jordan Elimination is

| 6  24  1  |      |  0.159 -0.778 0.508 |
| 13 16  10 |   =  |  0.011 0.159 -0.107 |
| 20 17  15 |      | -0.024 0.857 -0.490 |

I don't really know why is it different and not sure which is correct. Please help.

Answer 1

You must work in the integers modulo $26$, not over the reals (which is what your program seems to be doing).

To check the inverse, just do the multiplication: e.g. the inner product of the row $(6\,\,24\,\, 1)$ with the column $(8\,\,21\,\,21)$ equals $6 \times 8 + 24 \times 21 + 1 \times 21 = 573$, which modulo 26 equals 1 (as $572$ is divisible by $26$), so we get a $1$ at the left upper number of the product matrix. Similarly for the other ones.

You can do Gaussian elimination, but division works differently in the integers modulo $26$. (For one thing, you cannot always divide, as 26 has non-trivial divisors!) Look into working inside that ring first....