Number of solutions of a variable matrix

Question

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:

1. infinite solutions
2. one solution
3. 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,

Answer 1

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.

Answer 2

Hint: $$ (c \ne 0) \wedge (c \ne 1) \wedge (c \ne -1) $$