For what values of $k$ in this set of linear equations $Ax = b$ has no solutions, an infinite number of solutions and a unique solutions?
I know I want to be using Gaussian Elimination here, I've augmented the matrix
and I'm perfectly familiar with ERO's and back-solving for systems without unknown constants but this is new to me.
\begin{array}{ccc|c}
2 & 2 & 0 & 2\\
0 & k & 1 & 1\\
1 & 2 & k & 2
\end{array}
Would I try to be putting this into Row-Echelon form? I have an inkling by playing with it that $k = -1$ for no solutions and $k = 1$ for an infinite number of solutions. I can't do the Gaussian steps properly with a $k$ involved to produce some decent working though.
Thank you in advance for any help, solutions or tips. 🙂
Best Answer
Hint :
In order to have unique solutions, the determinant should be nonzero :
$$ \det(A) = 0 \Leftrightarrow \begin{array}{|ccc|c} 2 & 2 & 0 & \\ 0 & k & 1 \\ 1 & 2 & k \end{array} =0 \Leftrightarrow 2(k^2-1) = 0 \Leftrightarrow k = +-1$$
Now, by plugging $k=1$ to our matrix and doing a Reduced Echelon Form Transformation :
$$\left( \begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)$$
and by plugging $k=-1$, executing again a Reduced Echelon Form Transformation :
$$\left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right)$$
Can you now derive conclusions for inconsistent, unique solution and infinite solutions?