# Thread: Rank of matrices and system of linear equations

1. ## Rank of matrices and system of linear equations

Let be
$A=\left(\begin{array}{ccc}
1& 1 & 1 \\
1 & -1 & 3 \\
1 & 3 & -1
\end{array} \right),\: b=\left(\begin{array}{ccc}
-4 \\
0 \\
4 \end{array} \right)$
.

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!

2. $r(A)=2 \:\:\:\: r(A|b)=3 \:\:\:\:\:r(A)\ne r(A|b) \rightarrow$ there is no soultion over $\mathbb{R}$. You can define the ranks by Gaussian elimination.
Over GF(2) and GF(3) you should see the elements of the matrix modulo 2 and 3.
I think it can be solved over GF(2) and GF(3) (unique solution and infinitely many solutions - 2 free parameters).