## A system over Z7

Determine all solutions of a system of equations with the following extended matrix over $$\mathbb Z_7$$:

$$\left(\begin{array}{ccc|c} 3&5&0&1\\ 1&2&2&4\\ 1&3&2&3\\ \end{array}\right)$$

• #### Resolution

We first add twice the first row to both the second and the third row. Then we add three times the second row to the third.

$\left(\begin{array}{ccc|c} 3&5&0&1\\ 1&2&2&4\\ 1&3&2&3\\ \end{array}\right)\sim \left(\begin{array}{ccc|c} 3&5&0&1\\ 0&5&2&6\\ 0&6&2&5\\ \end{array}\right)\sim \left(\begin{array}{ccc|c} 3&5&0&1\\ 0&5&2&6\\ 0&0&1&2\\ \end{array}\right)$

In the backward substitution we need to divide by five, that is equivalent with the multiplication by three (and vice versa).

• #### Result

$$\mathbf x=(2{,}6,2)^T.$$