Number of solutions of a variable matrix Given matrix:
$$\left(\begin{array}{ccc|c}
c & c & 1-c &  1\\
c & c^{2} & 1-c^{2}  & 1\\
2c & c+c^{2} & 2-2c  & c+2
\end{array}\right)$$
row reduced to:
$$\left(\begin{array}{ccc|c}
c & 1 & 0  & 1\\
0 & c-1 & 1-c &  0\\
0 & 0 & c+1 &  1
\end{array}\right)$$
question asks to find the values of 'c' for which the matrix has:


*

*infinite solutions

*one solution

*no solution


my answer:
the column vectors are linearly independent thus the Rank[A|b] = Rank[A] = n = 3
then for all c ≠ 1 we have one solution
And for c = 1 we have no solutions.
Thanks,

let us look at the cases separately. 
in case $c = 0,$ you have the system $\pmatrix{0&0&1&|&1\\0&0&1&|&1\\0&0&2&|&2}$ which is and has infinitely many solutions.
in case $c = 1,$ you have the system $\pmatrix{1&1&0&|&1\\1&1&0&|&1\\2&2&0&|&3}$ which has no solutions.
in case $c \ne 0, c \neq 1$ we can reduce 
$\pmatrix{c&c&1-c&|&1\\c&c^2&1-c^2&|&1\\2c&c+c^2&2-2c&|&c+2} \rightarrow
\pmatrix{c&c&1-c&|&1\\0&c^2-c&c-c^2&|&0\\0&-c+c^2&0&|&c} \rightarrow
\pmatrix{c&c&1-c&|&1\\0&c^2-c&c-c^2&|&0\\0&0&c-c^2&|&c}
$ 
has a unique solution.
