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I am investigating the hill cipher. The determinant of my encryption matrix is a negative number, and I need to find the inverse modulo (using mod 26) of this determinant. This is eventually used to create the deciphering matrix. Do I take the absolute value of the determinant, and hence make it positive, or is there a method to find it as a negative.

The determinant to find reciprocal modulo 26 of is -441. Thanks for the help.

Bill Dubuque
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Jon-Luc Catinari
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  • By the inverse aw in the dupe: $-441\equiv 1\Rightarrow (-441)^{-1}\equiv 1^{-1}.\ $ Or use $,(ab)^{-1}\equiv a^{-1}b^{-1},$ for $,a\equiv -1\ \ $ – Bill Dubuque Mar 19 '24 at 07:49
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    I am in year 11, can you please help explain this a bit more i do not fully understand. Could you help by explaining the process of finding the inverse modulo of -441. – Jon-Luc Catinari Mar 19 '24 at 07:56
  • $-411+17\cdot26=1$ so $-441\equiv1 \bmod {26}$ and $(-441)^{-1}\equiv1^{-1} \equiv1 \bmod {26}$. – Jaap Scherphuis Mar 19 '24 at 09:47

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