If a system of linear equations has augmented matrix $$[A|\vec{b}]=\begin{bmatrix} 1 & a & 0 & 1\\ 0 & 1 & b & 1\\ c & 0 & 1 & 1 \end{bmatrix}$$ for which values a, b, c, does the system have a solution? Are there values with a unique solution? If so, give all the values.
So far I have found that c=0 for there to be a solution so that the Rank is 3 = # of equations. My guess is that a and b can be any value, as when I plug them in there is a solution, but apparently I'm supposed to list all values? That doesn't seem right.
For unique solution, we need$$ \begin{vmatrix} 1 & a & 0 \\ 0 & 1 & b \\ c & 0 & 1 \end{vmatrix} \neq 0$$ This holds if and only if $1+abc \neq 0$.
When $1+abc = 0$:
The system of equations can be written as: \begin{align*} x + ay &= 1 \\ y+bz &= 1\\ cx + z &= 1 \end{align*} Eliminating $z$, we have \begin{align*} x+ay &= 1\\ -bcx + y &= 1-b \end{align*} This has no unique solution and the condition for infinitely many solutions is $$-bc(1-ay) + y = 1 - b \Rightarrow -bc + (1+abc)y = 1-b \Rightarrow 1-b+bc = 0$$ In this case, $y$ can be arbitrary, $x = 1-ay$ and $z = 1- c(1-ay)$.