# Thread: system of linear equations with an unknown coefficient

1. ## system of linear equations with an unknown coefficient

Determine all values of k for which the system

10 k -1 0
k 1 -1 0
2 1 -3 0

has nontrivial solution.

I have written the augmented matrix, with the solutions on the fourth row. My first instinct was to use Gaussian elimination to reduce the matrix to echelon form and peal off the solutions. However, this quickly became impossibly complex. Can anyone offer me any advice?

2. Well, you have a homogeneous system, right? That means that if there is only one solution, it must be the trivial solution. Therefore, in order to have nontrivial solutions, you must have infinitely many solutions. What does that tell you about the matrix

$A=\begin{bmatrix}10 &k &-1\\ k &1 &-1\\ 2 &1 &-3\end{bmatrix}?$

3. My book seems too be directing me to reduce the system into row echelon form. Your hint leads me to believe that there may be an easier way, but I can't see it. Can you be more explicit?

4. Wait - would the possible values of k be the set of real numbers? Since there are an infinite number of possible nontrivial solutions?

5. What do you know about the determinant of a square matrix that admits infinitely many solutions?

6. Ah - I got it. Thank you for the help!

7. You're welcome. Have a good one!