I have this problem:
With this system, determine the values of $K$ to the system have:
a) One only solution b) Don`t have solution c) Infinite solutions
$x-3z=-3$
$2x+Ky-z=-2$
$x+2y+Kz=1$
How do I solve that system?, I already know the answer but I haven't could reach it.
Thank you!
First, we convert the system of equations to an augmented matrix (which is essentially just another way of writing the system of equations):
$$ \left[\begin{array}{ccc|c} 1 & 0 & -3 & -3 \\ 2 & K & -1 & -2 \\ 1 & 2 & K & 1 \\ \end{array}\right]. $$
We can think of $K$ as some number that we don't happen to know. We apply row operations until we get the matrix in reduced row echelon form (this is referred to as Gauss-Jordan Elimination).
Along the way, we might need to apply a row operation which amounts to dividing a row by $K$ (or some function of $K$). We can only do this if $K \neq 0$. We will therefore need to keep track of these assumptions, and resolve these cases separately.
Likely, most of the time, we'll reach a reduced row echelon form like the following: $$ \left[\begin{array}{ccc|c} 1 & 0 & 0 & ??? \\ 0 & 1 & 0 & ??? \\ 0 & 0 & 1 & ??? \\ \end{array}\right] $$ and the values where the question marks determine the $x,y,z$-values, and thus gives a unique solution.
It might happen (for some values of $K$) that we reach a reduced row echelon form like: $$ \left[\begin{array}{ccc|c} 1 & 0 & 0 & ??? \\ 0 & 1 & 0 & ??? \\ \color{blue} 0 & \color{blue} 0 & \color{blue} 0 & \color{blue}{\neq 0} \\ \end{array}\right] $$ which would imply no solutions (i.e., the system is inconsistent).
If, for some value of $K$, the system is consistent (i.e., there are solutions) and the reduced row echelon form has a row of zeros, there will be infinitely many solutions. E.g., it could look like $$ \left[\begin{array}{ccc|c} 1 & 0 & 0 & ??? \\ 0 & 1 & 1 & ??? \\ \color{blue} 0 & \color{blue} 0 & \color{blue} 0 & \color{blue} 0 \\ \end{array}\right]. $$
So your task is to perform the row operations to the augmented matrix, and determine which values of $K$ give rise to which type of reduced row echelon form.
(If doing this by hand, it's more likely than not that arithmetic errors arise; so try to keep on top of these.)