3 x 3 Matrices, and solving a series of equations.

[finchie_88]

I can solve equations using matrices (finding the inverse method etc), and in my textbook at the moment it sometimes gives questions where the determinant is zero,I understand what that means, and I can spot whether the equations are inconsistent, the same thing (just multiplied by a number), or several distinct lines, but sometimes, they form prisms. My question is: how can you tell from the equations that they form a prism?
Here is an example question from my book...
Solve:
$$\left(\begin{array}{ccc}1&1&1\\2&3&-4\\5&8&-13\end{array}\right)$$x$$\left(\begin{array}{c}x&y&z\end{array}\right)$$=
$$\left(\begin{array}{c}4&3&8\end{array}\right)$$

edit: Sorry about thedodge matrices, its my first time doing them.

 

[Fermat]

I was curious about this question, so ...

The eqns are,

x + y + z = 4 ------------------(1)
2x + 3y - 4z = 3 ---------------(2)
5x + 8y - 13z = 8 --------------(3)

I plotted these three eqns as planes and found that they do form a prism.
Then I noticed that the lines of intersection of the three planes all looked parallel.
I then took each pair of planes to find the line/plane of intersection.

x + y + z = 4 ------------------(1)
2x + 3y - 4z = 3 ---------------(2)

(2) - 2*(1) gives,

y - 6z = -5 -------------(4)
========

x + y + z = 4 ------------------(1)
5x + 8y - 13z = 8 --------------(3)

(3) - 5*(1) gives,

3y - 18z = -12
y - 6z = -4 -------------(5)
=========

2x + 3y - 4z = 3 ---------------(2)
5x + 8y - 13z = 8 --------------(3)

5*(2), 2*(3) gives,

10x + 15y - 20z = 15 ------------(6)
10x + 16y - 26z = 16 ------------(7)

(7) - (6) gives,

y - 6z = 1 -------------(8)
========

The lines (or planes) (4), (5) and (8) all have the same slope (y=6z), and so are all parallel.

So, I don't think you can tell if a set of eqns form a prism just by casual observation, but if you do a little work on them, to show that they the intersecting planes all have the same "slope", then that should show that these eqns form a prism.

HTH

 

[finchie_88]

Thank you for the help. It is greatly appreciated.

You said that you plotted the planes, did you do that using a computer? (If you didn't, you must have some good art skills).

 

[Fermat]

Yeah, I plotted them on my computer.
I used Autograph.

 

[0rthodontist]

Well, they will form a "prism" if any 2 rows of the augmented matrix are consistent with each other, but all 3 together are not. This will happen when there are exactly 3 pivot positions in the augmented 3x4 matrix, and one of the pivot positions is in column 4.