For the augmented matrix,

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 2 & { - 3} & a \\

2 & 3 & { - 2} & b \\

3 & 1 & {11} & c \\

\end{array}} \right)

$

After performing reduced row reduction as follows

r2=r2-2*r1

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 2 & { - 3} & a \\

0 & { - 1} & 4 & {b - 2a} \\

3 & 1 & {11} & c \\

\end{array}} \right)

$

r3=r3-3*r1

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 2 & { - 3} & a \\

0 & { - 1} & 4 & {b - 2a} \\

0 & { - 5} & {20} & {c - 3a} \\

\end{array}} \right)

$

r2=r2*(-1)

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 2 & { - 3} & a \\

0 & 1 & { - 4} & {2a - b} \\

0 & { - 5} & {20} & {c - 3a} \\

\end{array}} \right)

$

r3=r3+5*r2

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 2 & { - 3} & a \\

0 & 1 & { - 4} & {2a - b} \\

0 & 0 & 0 & {13a - 5b + c} \\

\end{array}} \right)

$

r1=r1-2*r2

$\displaystyle

\left( {\begin{array}{*{20}c}

1 & 0 & 5 & {b - 3a} \\

0 & 1 & { - 4} & {2a - b} \\

0 & 0 & 0 & {13a - 5b + c} \\

\end{array}} \right)

$

Would it be true to say the following:

1) There are no values of a,b,c that the equations have a unique solution.

2) That there are some values of a,b,c such that the equations have no solutions.

3)There are no values of a,b,c, that the equations have infinitely many solutions

4) There are no values of a,b,c, such as that {(1,2,3),(2,3,1),(-3,-2,11),(a,b,c)} are linearly independent.