How do I solve this system of linear equations using modular arithmetics

Question

How do I solve this system of linear equations using augmented matrices $\mod 7$:

$3x + 6y + 2z = 1 \\ 2x + 4y + z = 6 \\ 4x + y + 2z = 5 ?$

A hint is given that the first calculation for Row 1 is: $R1 = \alpha R1$, where $3 \alpha = 1 \pmod 7$.

Answer 1

Hint:

It is done by row reduction as usual, but you have to compute first the inverses modulo $7$. This is given by the following relations: $$ 2\cdot 4=1,\quad 3\cdot5=1,\quad 6^2=1.$$

Then one should make each successive pivot equal to $1$, multiplying each row with the inverse of the element in the same column as the pivot. This is how it goes: $$\color{red}{\begin{matrix}{}\times 5\\{}\times4 \\{}\times2 \end{matrix}}\begin{bmatrix} 1&2&4&5\\1&2&4&3\\ 1&2&4&4\end{bmatrix}$$ Thus, the system has no solution.