anonymous
  • anonymous
For which values of a will the following system have no solutions? Exactly one solution? Infinitely many solutions? x+6y+4z=2 4x+25y+31z=7 2x+13y+(7+a^2)z=-1+a I tried elimination but that didn't work... I don't know how I would use matricies for this either.
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
anonymous
  • anonymous
First, turn it into a matrix!
anonymous
  • anonymous
Umm okay...

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anonymous
  • anonymous
Okay.
anonymous
  • anonymous
Here do you understand how I got this? [1 6 4 2 ] [4 25 31 7 ] [2 13 7a^2 -1+a]
anonymous
  • anonymous
Yeah I do.,
anonymous
  • anonymous
Actually it should be 7+ a^2 so [1 6 4 2 ] [4 25 31 7 ] [2 13 7+a^2 -1+a] OK now do you know how to row reduce this?
anonymous
  • anonymous
Yeah but what about that a?
anonymous
  • anonymous
anonymous
  • anonymous
We're going to work with that later. It shouldn't get in the way because it's in the bottom row.
anonymous
  • anonymous
I have to still manipulate it like normal correct?
anonymous
  • anonymous
Yes. So first try to get 0s in the first column
anonymous
  • anonymous
|dw:1358201674350:dw|
anonymous
  • anonymous
No wait. That's not right.
anonymous
  • anonymous
Nope :P OK I'll give you the first step. Subtract 4 times the first row from the second row.
anonymous
  • anonymous
Sorry about that. This Linear algebra is fun but totally new to me.
anonymous
  • anonymous
Nono don't be sorry, it's not an easy course at all! I just took it last semester so it's still fresh in my mind. Everybody said it would be easier than calculus, but I think they were crazy, linear is way harder...
anonymous
  • anonymous
If we went 4* Row 1 we get -3 in the 2nd row for the first term.
anonymous
  • anonymous
OK so subtracting 4*R1 from R2 you get. 4*R1 = 4 * [ 1 6 4 2 ] = [4 24 16 8] R2 - 4*R1 = [ 4 25 31 7 ] - [4 24 16 8] = [0 1 15 -1] Make sense? So now your matrix looks like [1 6 4 2 ] [0 1 15 -1 ] [2 13 7+a^2 -1+a]
anonymous
  • anonymous
Wait... Oops.
anonymous
  • anonymous
KK. Let me try myslef again.
anonymous
  • anonymous
I was looking at another Matrix >.< .
anonymous
  • anonymous
Kk. I got what you got. Now I would go and divide Row 3 by 2 right?
anonymous
  • anonymous
anonymous
  • anonymous
Nope you would do Row 3 - 2 times Row 1. You want to get a 0 in the first spot of row 3
anonymous
  • anonymous
Okay, that works too.
anonymous
  • anonymous
For row 3 I got [0 1 5+a^2 -3+a]
anonymous
  • anonymous
Wait, third colums in wrong.
anonymous
  • anonymous
[0 1 -1+a^2 -3+a]
anonymous
  • anonymous
Perfect very good! :D [1 6 4 2 ] [0 1 15 -1 ] [0 1 -1+a^2 -3+a] almost there. Now you want to get a 0 in row 3 column 2
anonymous
  • anonymous
[0 1 -1+a^2 -5+a] . Woudn't it be that actually?
anonymous
  • anonymous
Oops yes! Missed that, just skimmed over it. Good job noticing that. So now it's [1 6 4 2 ] [0 1 15 -1 ] [0 1 -1+a^2 -5+a]
anonymous
  • anonymous
K I got [0 0 -16+a^2 -4+a]
anonymous
  • anonymous
Perfect! OK I have to go so I'll just go over the answer and stop giving you hints. If you want to guess for yourself do that first then read my answer! [1 6 4 2 ] [0 1 15 -1 ] [0 0 a^2-16 a-4] is equal to (by difference of squares) [1 6 4 2 ] [0 1 15 -1 ] [0 0 (a+4)(a-4) a-4] So you've got (a+4)(a-4) * z = a-4 If the left equals 0, but the right equals a number, then there is no solution because: 0*z=1 has no solution. So when does the left equal 0 but the right equal a number? When a = -4. Then 0*z = -8, no solution. If they both equal a number, let's say a=5 then [1 6 4 2 ] [0 1 15 -1 ] [0 0 9 1 ] You now have 3 pivot columns, which means that there is only 1 solution. If they both equal 0, then you will have a free variable in the third column and you will have infinite solutions: [1 6 4 2 ] [0 1 15 -1 ] [0 0 0 0 ] When do they both equal 0? When a =4, infinite solutions. Good luck in linear algebra! It's a fun class, but make sure you don't fall behind, it gets really hard!!
anonymous
  • anonymous
If you've got any questions about my explanation, ask them and I'll answer later.
anonymous
  • anonymous
WHat's a Pivot column?
anonymous
  • anonymous
And thanks for the encouragement :) .

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