A system over Z7
Task number: 2480
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
\(\mathbf x=(2{,}6,2)^T.\)
