Let be

$\displaystyle A=\left(\begin{array}{ccc}

1& 1 & 1 \\

1 & -1 & 3 \\

1 & 3 & -1

\end{array} \right),\: b=\left(\begin{array}{ccc}

-4 \\

0 \\

4 \end{array} \right) $.

What is the rank of A and [A|b] expanded matrix over GF(2), GF(3) and (where GF(2) and GF(3) are the Galois fields of two/three elements and R means real numbers).

What can we say we about the solvability of Ax=b system of linear equations in the three cases (GF(2), GF(3), R)?

I would be very grateful, if you could help me!

Thank you!