Finding k so a matrix system has no solution

[danizh]

Please help me solve this problem associated with matrices; I can't seem to figure it out.

The following reduced augmented matrix represents a system of equations of three plans:

1 0 0 | 3
0 1 0 | 4
0 0 (k^2)-k | k

For what value(s) of k will this system have no solution?

 

[TD]

If the z-coëfficiënt becomes 0 while the constant isn't 0, your system won't have a solution.

So $$k^2 - k = 0 \Leftrightarrow k = 0 \vee k = 1$$.

But for k = 0, the constant is 0 as well so the last equation is no longer independent, then you have a 2x3 system (which has infinite solutions).

For k = 1, it has no solution

 

[danizh]

Awesome, thanks for the help!
Wouldn't "+1" have no solution as well?

 

[TD]

I mean +1 lol, it has a solution for -1

I'll correct.

---

Done.

Since, for k = -1, the last row becomes: 0 0 2 | 1 => 2z = 1 => z = 1/2

 

[danizh]

But (-1)^2 - 1 = 0 as well. So there are no solutions for "+/- 1," I suppose.
Please correct me if I am wrong.

 

[TD]

Watch out, mind your minus! For k = -1, you get:

$$k^2 - k \Rightarrow \left( { - 1} \right)^2 - \left( { - 1} \right) = 1 + 1 = 2$$

Which is what I wrote at the end of my previous post.

 

[danizh]

Sorry, I missed that.
Thanks for the help!

 

[TD]

No problem

 

[danizh]

I don't think that's true, since anything greater than 2 seems to have no solution as well.

 

[TD]

I don't think that's true, since anything greater than 2 seems to have no solution as well.

How is that?

Take k = 3, the last row becomes: 0 0 6 | 3 => 6z = 3 <=> z = 1/2