0
$\begingroup$

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.


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 $var_1=var_2=var_3=0$.

$\endgroup$

1 Answer 1

1
$\begingroup$

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

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

Does this answer your question?

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.