I am completing a practice questions sheet for the topic "systems of linear of equations" and I'm having trouble on one of the questions.
1. Consider the system of equations $$\begin{aligned} x + 2y - z &= -3 \\\ \end{aligned}$$ $$\begin{aligned} 3x + 5y + kz &= -4 \\\ \end{aligned}$$ $$\begin{aligned} 9x + (k+13)y + 6z &= 9 \\\ \end{aligned}$$ a) Express these equations as an augmented matrix
which I think is: $$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 3 & 5 & k & -4 \\ 9 & (k+13) & 6 & 9 \end{array}\right] $$
b) Show that this matrix can be row-reduced to
$$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & -3 \\ 0 & 1 & -k-3 & -5 \\ 0 & 0 & k^2-2k & 5k+11 \end{array}\right] $$
c) hence answer the following questions
for what value(s) of k does the system have
. no solutions: I think when k=0, k=2 and k does not = -11/5
. a unique solution: when k= any real number other than 2,0 and -5/11
. infinitely many solutions: I couldn't find anything for this so I believe that this system doesn't have infinite solutions (correct me if I'm wrong)
the question that I'm particularly stuck on is:
"Each of these equations represents a plane. In each case in (the above question), give a geometric description of the intersection of the three planes."
You are correct in everything you said — indeed the matrix cannot have infinitely many solutions. Sometimes questions are like that! They ask for “how many” and the answer ends up being $0$.
Onto the second part, if you imagine two lines in the plane, they are either parallel or intersect. With planes in space, it is the same thing:
If any two planes are parallel, then there is no triple point of intersection.
It is possible that all of the planes pairwise intersect, but there is no triple intersection point (think of three faces in a triangular prism).
More often than not, the planes don’t have any parallel stuff happening and they all pass through the same point
Sometimes all three planes pass through a line, in which case you have infinitely many solutions
Final case is when they are all just the same plane, so you get infinitely many solutions again