I had equations and expressed them as an augmented matrix. Shown below:

$$ \left[ \begin{array}{ccc|c} 1&2&-1&-3\\ 3&5&k&-4\\ 9&k+13&6&9\\ \end{array} \right] $$

I then row reduced the system to this.

$$ \left[ \begin{array}{ccc|c} 1&2&-1&-3\\ 0&1&-k-3&-5\\ 0&0&k^2-2k&5k+11\\ \end{array} \right] $$

I found out that it has no solutions when $k = 0$ and $k=2$, a unique solution for any real number other than $0$ and $2$ and that it isn't possible to have infinitely many solutions.

The very last part of the problem asks:

(i) Each of these equations represents a plane. In each case (no solutions, a unique solution and infinitely many solutions) give a geometric description of the intersection of the three planes.

I am stuck on this bit. Any help appreciated.

  • $\begingroup$ How can the intersection of three planes give no solution? (Hint: when does the intersection of two or more lines have no solution?) $\endgroup$ – videlity Feb 22 '18 at 4:38
  • $\begingroup$ I was under the impression that if all values on the last row of the matrix are 0 expect for the very last i.e. after the line, then the system is inconsistent and has no solutions. For this, I found out what the values of k could be in k^2 - 2k = 0 and got 0 and 2. So it does have solutions, even if k = 2 or k = 0? $\endgroup$ – Juan Pablo Feb 22 '18 at 5:02
  • $\begingroup$ Yes, that's the algebraic interpretation. Your working out is correct. There's no solution for $k=0$ or $k=2$. But the question is asking for a geometric description. I see you have answered this in your other comment. $\endgroup$ – videlity Feb 22 '18 at 5:16

When $k=0$, none of the planes are parallel to each other. Any two of them intersect at a line but three of them do not intersect together.


When $k=2$, the second and third equation have their left hand side being multiple of each other, try to interpret that.

  • $\begingroup$ Ok to help me understand it better, I graphed the equations. When k = 0, I can see that what you said makes sense. The planes intersect each other but do not intersect together. I see now that the left hand side of the second and third equations is a multiple of each other. So the third equation has been scaled up and is now parallel to the second equation. $\endgroup$ – Juan Pablo Feb 22 '18 at 5:13
  • $\begingroup$ yup, just like parallel line do not intersect, parallel planes that are not identical do not intersect as well. $\endgroup$ – Siong Thye Goh Feb 22 '18 at 5:15
  • $\begingroup$ OK, I think I now understand how to describe geometrically the intersection of the planes when $k = 0$ and $k = 2$ but what about when $k$ does not equal those values, e.g. $k = 1$. I know that this would make it consistent with a unique solution. I've graphed it up and it looks similar to what it looks like when $k=0$, none of the planes are parallel and they all intersect but don't intersect together? It's pretty much the same answer??? $\endgroup$ – Juan Pablo Feb 22 '18 at 5:25
  • $\begingroup$ by gaussian elimination, you have proven that they intersect at a single point. this page should help you in visualizing. $\endgroup$ – Siong Thye Goh Feb 22 '18 at 5:35
  • $\begingroup$ I don't really understand that. So by gaussian elimination, I have proved that when $k\neq0$ and $k\neq2$ all planes intersect at a single point? $\endgroup$ – Juan Pablo Feb 22 '18 at 5:43

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