How do I find for which K parameter we have zero solutions, single solution and infinite number of solutions? I can't seem to do it with the Gaussian Elimination.
Hello, GIPC!
Here's a start . . .
Consider the determinant of the system: . $\displaystyle D \;=\;\begin{vmatrix}1 & k^2-6 & 4(k+1) \\ \text{-}1 & 2(k^2-3) & 4(k+1) \\ 0 & 3(k^2-4) & 6(k+1) \end{vmatrix}$
$\displaystyle \begin{array}{ccccccc}
x & + & (k^2-6)y &+& (4k+4)z &=& 5k+3 \\
\text{-}x &+& (2k^2-6)y &+& (4k+4)z &=& 6k+3 \\
& +&(3k^2-12)y &+& (6k+6)z &=& 9k+6 \end{array}$
Find for which parameter $\displaystyle k$ we have zero solutions, single solution and infinite number of solutions.
Factor $\displaystyle 2(k+1)$ from column 3: . $\displaystyle D \;=\;2(k+1)\begin{vmatrix}1 & k^2-6 & 2 \\ \text{-}1 & 2(k^2-3) & 2 \\ 0 & 3(k^2-4) & 3 \end{vmatrix}$
Evaluate: .$\displaystyle D \;=\;6(k+1)\bigg[\left[2(k^2-3) - 2(k^2-4)\right] + \left[(k^2-6) - 2(k^2-4)\right] + 0\,\bigg] $
And we have: . $\displaystyle D \;=\;6(k+1)(-2k^2+8) \;=\;-12(k+1)(k^2-4) \;=\;-12(k+1)(k-2)(k+2)$
. . Hence: .$\displaystyle D \:=\: 0\,\text{ for }\,k \:=\:\text{-}1,\: 2,\:\text{-}2$
Can you take it from here?
Are you not trying anything yourself? For example, it shouldn't take long to see that you can reduce the second equation to $\displaystyle (k^2- 1)y+ 2(k+1)z= 3k+ 2$. And, as long as $\displaystyle k\ne \pm 1$, we must have $\displaystyle y= \frac{3k+ 2- 2(k+1)z}{k^2- 1}$ for any value of z. And then $\displaystyle x= 5k+3- (4k+4)z- (k^2- 6)y$.