In 'Algebra', Artin writes that the system of equation:
$$8x+3y = 3$$ $$2x+6y = -1$$
have no solutions in $\mathbb{F}_2$ and $\mathbb{F}_3$ as the determinant (of the coefficient matrix) evaluates to $42$ which is equal to $0$ (mod $2$, $3$ and $7$).
However he mentions that there happens to be a solution $(3, 0)$ in $\mathbb{F}_7$ though the determinant is zero modulo $7$. However this solution is not evident from the non-zero determinant (invertible matrix) rule we generally look at to determine whether solutions exist or not.
Should we always reduce the coefficient matrix elements (modulus $p$) before evaluating value of determinant to check if a solution exists in $\mathbb{F}_p$? Is there a better way to check for existence of solutions in prime fields?
It is evident from the matrix. in the field $Z/7Z,$ the system is $$ x+3y = 3, $$ $$ 2x+6y = 6. $$
Solutions are $(3,0), \; \; $ $(0,1), \; \; $ $(4,2), \; \; $ $(1,3), \; \; $ $(5,4), \; \; $ $(2,5), \; \; $ $(6,6). \; \; $