Solve 4 equations with a parameter using matrices

Question

Find for what values of the parameter $k$ there are:

1. A single solution.
2. Infinite solutions.
3. No solutions.

$x_1 +x_2 +2x_4 = 1 $

$x_1 +2x_2 +2x_3 +2x_4 = 3$

$x_1 +2x_2 +4x_3 +(2-k)x_4 = 3$

$x_1 +x_2 +(k+3)x_4 = k^2$

I put all these equations in a matrix and made it canonical

$\left( \begin{matrix}1 & 0 & 0 & 0 & k^3-2k^2-3k+5 \\ 0 & 1 & 0 & 0 & -k^3+3k+2\\ 0 & 0 & 1 & 0 & \frac{k^3-3k}{2} \\ 0 & 0 & 0 & 1 & k^2-3\end{matrix}\right)$

What do I do now?

Answer 1

You shouldn't go all the way down to reduced row echelon, because you may be dividing by zero depending on what $k$ is. So I reduced it down to\begin{bmatrix} 1 & 1&0&2&1\\ 0&1&2&0&2\\ 0&0&2&-k&0\\ 0&0&0&k+1&k^2-1 \end{bmatrix}

Now from here it should be easy to see for what values of $k$ you have solutions.