How to determine value of a matrix

Question

I have this matrix. How do i determine the value of k such that the solution is consistent and compute the solution ?

2x + ky + 2z = 0

x − y + z = 1

y − z = k

I augment the matrix to get

2+k+2=0

1-1+1=1

0+1-1=k

Should i then compute the row reduced echelon form ? Can somebody show me the steps and consequently how to arrive to a solution k that is consistent? Thank you

Answer 1

Let consider the augmented matrix

$$\left[\begin{array}{ccc|c} 2& k& 2& 0\\ 1& -1& 1& 1\\ 0& 1& -1& k \end{array}\right]$$

then by the RREF obtain the condition on k for the system to be consistent, that is

$$\to \left[\begin{array}{ccc|c} 2& k& 2& 0\\ 0& -2-k& 0& 2\\ 0& 0& 2+k& k(-2-k)-2 \end{array}\right]$$

from which we can see that we need $k\neq -2$ in order to have solutions.

Answer 2

Empirically:

Add the second and third equations to get $x=k+1$. Now add twice the third to the first and substitute $x$:

$$2(k+1)+ky+2y=2k$$ or

$$y=-\frac2{k+2},$$ provided $k\ne-2$.

Then $z=y-k$ and $x=\dfrac12(-ky-2z)$ are well defined.