Reducing a matrix with Gaussian elimination

Question

How can I reduce the following matrix with Gaussian elimination? $$\begin{vmatrix} k & 0 & 1\\ 2 & 1- k & 2 \\ 1 & 2 & -k\\\end{vmatrix}$$

NB: The system is homogenous.

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Own attempt

I used the third row and multiplied and added/subtracted from 3rd and 2nd row. Unfortunately, I can't seem to figure out how to proceed from here. $$\begin{vmatrix} 0 & -2k & 1+k^2\\ 0 & -(3+k) & 2+2k \\ 1 & 2 & -k\\\end{vmatrix}$$

If it's interesting: I'm doing this so I can find out values on $k$ where the system only has the trivial solution $var_1=var_2=var_3=0$.

Answer 1

Mutltiply the first row by $\frac{2}{k}$ and subtract it from the second row, and multiply the first row by $\frac{1}{k}$ and subtract it from the third row to obtain

$$ \left [\begin{array}{ccc} k & 0 & 1 \\ 0 & 1 -k & 2- \frac{2}{k} \\ 0 & 2 & -\frac{1}{k} - k \\ \end{array} \right ]. $$

From this matrix, multiply the second row by $\frac{2}{1-k}$ and subtract it from the third row to obtain

$$ \left [ \begin{array}{ccc} k & 0 & 1 \\ 0 & 1-k & 2 - \frac{2}{k} \\ 0 & 0 & \frac{3}{k} - k \\ \end{array} \right ]. $$

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