# Who to solve this linear modular equation system?

I have this equation system:

a + b + c (mod 11) = 8

9a + 3b + c (mod 11) = 2

16a + 4b + c (mod 11) = 9

Unfortunately I totally don't know how to solve it. It is in general part of Lagrange's threshold scheme. I tried to search internet but found solutions only for one variable. On one website it was written this can be solved using matrix, but no example was given. Please give hints and general solution for that kind of equation systems. Thanks in advance.

• In this thread many of us discuss the approaches to inverting a matrix modulo a prime number. Solving a linear system modulo a prime number is essentially the same process with augmented matrices - hopefully familiar to you from a course in linear algebra. Try to absorb that! I see that you are studying crypto. Learning to think in terms of fields other than real (or complex) numbers is useful to you anyway in that endeavour, so take a look! – Jyrki Lahtonen Feb 19 '15 at 23:11
• @amzoti your solution is probably correct but I would like how to solve it myself – TN888 Feb 19 '15 at 23:24

The integers modulo $11$ form a field. So you can solve this system, essentially, just as you would for a system of equations in real variables. If you know how to solve equations by matrix methods, you can do the same here: start with $$\pmatrix{1&1&1&8\cr 9&3&1&2\cr 16&4&1&9\cr}\ .$$ Now I'll take the second row plus twice the first, remembering that $9+2\equiv0\pmod{11}$, and the third row minus $5$ times the first: $$\pmatrix{1&1&1&8\cr 0&5&3&7\cr 0&-1&-4&2\cr}\ .$$ For convenience I'll take the negative of the third row and swap it with the second, $$\pmatrix{1&1&1&8\cr 0&1&4&-2\cr 0&5&3&7\cr}\ .$$ Third row minus $5$ times the second: $$\pmatrix{1&1&1&8\cr 0&1&4&-2\cr 0&0&5&6\cr}\ .$$ Now the third equation says $5c\equiv6\pmod{11}$ and hence $c=10=-1$.
See if you can finish the job yourself. You should get $b=2$ and $a=7$.