$$\text{Let} \ \ A= \left[ \begin{array}{cccc} 2&-1&3&4\\ 2&c&0&6\\ -2&1&c-5&-4\\ 2&c&0&c+5\\ 0&c+1&c-5&2 \end{array} \right], \ \ \ b = \left[ \begin{array}{c} 1\\ 1\\ -1\\ 2\\ 0 \end{array} \right], \ \ \text{and} \ \ X = \left[ \begin{array}{c} x\\ y\\ z\\ w \end{array} \right] $$
Determine all values of c for which the system $AX = b$ has
(i) no solution,
(ii) infinitely many solutions,
(iii) unique solution.
Write the solutions in cases (ii) and (iii).
Firstly, I have written it as an augmented matrix.
After that I have tried to find echelon form. I got one bad row.
So can't solve this question. Please help me.