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Math Help - Rank of matrices and system of linear equations

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    43

    Rank of matrices and system of linear equations

    Let be
    A=\left(\begin{array}{ccc}<br />
1& 1 & 1 \\<br />
1 & -1 & 3 \\<br />
1 & 3 & -1<br />
\end{array} \right),\:  b=\left(\begin{array}{ccc}<br />
-4 \\<br />
0 \\<br />
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!
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  2. #2
    Junior Member
    Joined
    Oct 2010
    Posts
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    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).
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