UsukiDoll
  • UsukiDoll
solve the system of linear equations using the Gauss–Jordan elimination method. 2x-3y=8 4x+y=-2
Mathematics
schrodinger
  • schrodinger
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UsukiDoll
  • UsukiDoll
|dw:1361067093589:dw|
UsukiDoll
  • UsukiDoll
|dw:1361067118822:dw|
UsukiDoll
  • UsukiDoll
ugh I missed that one on my paper! that subtraction because -16-(-2) is - 14

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UsukiDoll
  • UsukiDoll
anyway|dw:1361067211535:dw|
UsukiDoll
  • UsukiDoll
|dw:1361067247174:dw|
UsukiDoll
  • UsukiDoll
|dw:1361067302376:dw|
Callisto
  • Callisto
|dw:1361067336310:dw|
UsukiDoll
  • UsukiDoll
|dw:1361067340708:dw|
Callisto
  • Callisto
|dw:1361067371987:dw|
UsukiDoll
  • UsukiDoll
I'm using elementary row operations.
UsukiDoll
  • UsukiDoll
oh crud
UsukiDoll
  • UsukiDoll
spotted anotehr error. not my day today
UsukiDoll
  • UsukiDoll
|dw:1361067436003:dw|
UsukiDoll
  • UsukiDoll
arghhh...
UsukiDoll
  • UsukiDoll
ok really...I put this in reduced row echleon form. got my solution, but geez doesn't even equal back into the equation
Callisto
  • Callisto
Hmm... Try to do it again...
Callisto
  • Callisto
Since you've made a mistake in the first operation, it's hard to get the right answer :|
UsukiDoll
  • UsukiDoll
something is wrong with the y
UsukiDoll
  • UsukiDoll
ok let's redo this
UsukiDoll
  • UsukiDoll
|dw:1361067604121:dw|
UsukiDoll
  • UsukiDoll
augmented matrix [a b]
UsukiDoll
  • UsukiDoll
|dw:1361067630686:dw| triangles indicate main diagonal
UsukiDoll
  • UsukiDoll
so I have to get rid of the 4 and the -3.
UsukiDoll
  • UsukiDoll
2row 1 - row 2 --> row2
Callisto
  • Callisto
Hmm.. My usual practice is -2R1 + R2 -> R2
UnkleRhaukus
  • UnkleRhaukus
do you mean \[R_2\to R_2-2R_1\] @UsukiDoll
UsukiDoll
  • UsukiDoll
yes
UsukiDoll
  • UsukiDoll
|dw:1361067762579:dw|
UsukiDoll
  • UsukiDoll
|dw:1361067803078:dw|
UsukiDoll
  • UsukiDoll
so now... to get rid of the -3... 7row1 -3row2 --->row1
Callisto
  • Callisto
Why not do this: -1/7 R2 -> R2
UsukiDoll
  • UsukiDoll
achhhhhhhhhhhh
UsukiDoll
  • UsukiDoll
I don't wanna deal with fractions
Callisto
  • Callisto
But the answers are fractions :P
UnkleRhaukus
  • UnkleRhaukus
|dw:1361067930957:dw|
UsukiDoll
  • UsukiDoll
arghhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh I feel like Rarity Belle now
Callisto
  • Callisto
|dw:1361067978329:dw|
UsukiDoll
  • UsukiDoll
oh wait. sign errors everywhere. Is that why I am getting frustrateD?
UsukiDoll
  • UsukiDoll
wait a sec something is wrong already with row 2
UsukiDoll
  • UsukiDoll
|dw:1361068119755:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068152547:dw|
Callisto
  • Callisto
Relax~~~ |dw:1361068054013:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068165204:dw|
UsukiDoll
  • UsukiDoll
argh there could be so many combinations with this thing!
Callisto
  • Callisto
|dw:1361068179917:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068217096:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068228254:dw|
Callisto
  • Callisto
Just stop for a while, if you don't mind?
UsukiDoll
  • UsukiDoll
arghhhhhhhhhhhhh doing that got me x = 7
Callisto
  • Callisto
This is basically what you do for the first row operation -2R1 + R2 -> R2. I split it into two steps so that you can catch up the the arithmetic easier (I hope..) |dw:1361068290374:dw|
UsukiDoll
  • UsukiDoll
big numbers -_-
UsukiDoll
  • UsukiDoll
that means to get rid of the 6 I have to do 7r1 - 6r2
Callisto
  • Callisto
No?!
UsukiDoll
  • UsukiDoll
or -7r1+6r2
UsukiDoll
  • UsukiDoll
-.-
UsukiDoll
  • UsukiDoll
|dw:1361068464895:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068472375:dw| main diagonal in triangles
UsukiDoll
  • UsukiDoll
|dw:1361068495592:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068548208:dw|
Callisto
  • Callisto
Yup..Do you want to deal with big numbers or fractions?
UsukiDoll
  • UsukiDoll
|dw:1361068571208:dw|
UsukiDoll
  • UsukiDoll
I don't want to deal with fractions
UsukiDoll
  • UsukiDoll
|dw:1361068639980:dw|
Callisto
  • Callisto
Yes!!!
UsukiDoll
  • UsukiDoll
|dw:1361068650303:dw|
UsukiDoll
  • UsukiDoll
|dw:1361068693152:dw|
UsukiDoll
  • UsukiDoll
ok why the heck am I getting different answers despite the fact that I got it into reduced row echleon form. that's just...grrrrrrrrrrrrrrrrrr
UsukiDoll
  • UsukiDoll
just earlier I got x = -1 and y = -2
Callisto
  • Callisto
Hmm.. Careless mistakes perhaps?!
UsukiDoll
  • UsukiDoll
now x = 1/7 y = -18/7
UsukiDoll
  • UsukiDoll
what!
Callisto
  • Callisto
The one x=1/7; y=-18/7 is right :\
UsukiDoll
  • UsukiDoll
-.- is there more than one answer to this? There could be many many things that I could have done here.
Callisto
  • Callisto
No.
UnkleRhaukus
  • UnkleRhaukus
x = 1/7 y = -18/7 \(\LARGE\checkmark\)
Callisto
  • Callisto
There are only 3 cases for system of linear equations.. 1. No solution 2. Exactly one solution 3. Infinitely many solution.
UsukiDoll
  • UsukiDoll
I had a truckload
UnkleRhaukus
  • UnkleRhaukus
A quick string of row operations \[R_2\to R_2-2R_1\\R_2\to R_2/7\\ R_1\to R_1+3R_2\\R_1\to R_1/2\], if you have gotten a different solution you must have a made a mistake somewhere, it is very easy to make mistakes using Gauss–Jordan elimination .
Callisto
  • Callisto
1. No solution with a row that has all zero entries on the left, but non-zero entry on right => system is inconsistent 2. Exactly one solution After elementary row operations, no. of rows = no. of unknowns (/columns) => system is consistent 3. Infinitely many solution. After elementary row operations, no. of rows < no. of unknowns => system is consistent
UsukiDoll
  • UsukiDoll
but...yeah I suppose because I have seen some matrices that had a row of zeros on the left but with a nonzero on the right that is an inconsistent system .. How is it possible to make a mistake when it's so hard to spot one? I mean...I could've used different operations on the matrix. The combinations are everywhere.
UsukiDoll
  • UsukiDoll
Gauss Jordan is worse than Gaussian
UsukiDoll
  • UsukiDoll
ohhhhh what happens if I did Gaussian and then Gauss Jordon
UsukiDoll
  • UsukiDoll
because the answers to the Gaussian are supposed to be the same as the Gauss Jordon
Callisto
  • Callisto
Gaussian Jordan = Gaussian then Jordan lol!!!
UsukiDoll
  • UsukiDoll
but sometimes when I put the matrix into echleon form I can see the steps to making it row reduced
UsukiDoll
  • UsukiDoll
|dw:1361069849783:dw|
Callisto
  • Callisto
Guassian told you to make the left an upper triangular matrix Jordan asked you to further make the left an identity matrix
Callisto
  • Callisto
*Gaussian
UsukiDoll
  • UsukiDoll
|dw:1361069869054:dw|
UsukiDoll
  • UsukiDoll
|dw:1361069916982:dw|
UsukiDoll
  • UsukiDoll
-____________________- this is just Gaussian... WHAT THEEEEE!!!!!!! I see y = -18/7 oh I haven't solved yet oops
UsukiDoll
  • UsukiDoll
|dw:1361070002077:dw|
Callisto
  • Callisto
Just a minute..
UsukiDoll
  • UsukiDoll
|dw:1361070023727:dw|
UsukiDoll
  • UsukiDoll
uighhhhhhhhhhhhhhhh mind blown
UsukiDoll
  • UsukiDoll
Gaussian and then Gaussian Jordan yeahhhhh
UsukiDoll
  • UsukiDoll
I found the x = 1/7 just doing the Gaussian
Callisto
  • Callisto
|dw:1361070037568:dw|
UsukiDoll
  • UsukiDoll
yeahhhh...now I really should've done Gaussian first and then the Jordan version
Callisto
  • Callisto
Gaussian -> Make the left an upper triangular matrix Jordan -> Make the left an identity matrix Gaussian-Jordan => Make it an upper triangular matrix, then further make it an identity matrix. That's what Gaussan-Jordan is! Jordon helps you to get the answer by matching the left and the right That is for the row [1 0 | 1/7], you can tell immediately that x = 1/7 It's just the same as doing Gaussian elimination then back substitution.
UsukiDoll
  • UsukiDoll
dang that means I have to redo some practice problems...gawd not again wasting paper here.
UsukiDoll
  • UsukiDoll
oh yeah now I got it
UsukiDoll
  • UsukiDoll
yup works. I went way too ahead... x.x
UsukiDoll
  • UsukiDoll
so now I gotta correct all of these problems and write a proof...nice X___X
UsukiDoll
  • UsukiDoll
Callisto do you know how to write proofs? I got a partial first draft and I was wondering if you could critique it?
Callisto
  • Callisto
Are you doing high school maths or college maths?
UsukiDoll
  • UsukiDoll
college math
Callisto
  • Callisto
Proofs of??
UsukiDoll
  • UsukiDoll
like the one I am currently writing is a contradiction proof. contradicts a theorem at least 2-3 times
UsukiDoll
  • UsukiDoll
umm should I type it?
Callisto
  • Callisto
I.. didn't know what was a proof :S Please type it, if you don't mind!
Callisto
  • Callisto
*that instead of what
UsukiDoll
  • UsukiDoll
okkk... Let A and B be n x n matrices. Show that if AB is nonsingular then A and B must be nonsingular.(Hint Use Theorem 2.9) Theorem 2.9 states that the homogeneous system of n linear equations in n unknowns Ax = 0 has a nontrivial solution if and only if A is singular.
UsukiDoll
  • UsukiDoll
There's a contradiction....namely because if Matrix A is singular, then the inverse of Matrix A does NOT exist which means that AB doesn't exist as well.
UsukiDoll
  • UsukiDoll
It also has something to do with the homogenous system as well.
UsukiDoll
  • UsukiDoll
because basically a homogeneous system is always consistent. However, a homogeneous system's solution is always 0 and it's a trivial solution. Nontrivial solution in a homogenous system means that the solution IS NOT 0 at all!
UsukiDoll
  • UsukiDoll
so, what is given before proving is that A and B are n x n matrices A and B must be nonsingular AB is also nonsingular Theorem 2.9
UsukiDoll
  • UsukiDoll
If a matrix is nonsingular, an inverse exists...so Theorem 2.9 doesn't work at all
Callisto
  • Callisto
Forgive me, I don't understand the part "... which means that AB doesn't exist as well" I understand if A is singular, A^(-1) doesn't exist, but not the which means part..
UsukiDoll
  • UsukiDoll
oh yeah this is just a rough draft of it
UsukiDoll
  • UsukiDoll
hmm if A is nonsingular, an inverse exists. I remember back from Theorem 1.5 that it was AB=BA=In. B is the inverse of A. hmmm..if A is singular, yes the inverse doesn't exist. Let's have B = the inverse of A. Matrix A is singular Inverse doesn't exist so B doesn't exist AB is impossible to achieve unless A is nonsingular
Callisto
  • Callisto
Hmm.. \[AA^{-1} = A^{-1}A=I\]provided that \(A^{-1}\) exists... ------------- But what about \(B \ne A^{-1}\)? And we also have to show that B is also non-singular?!
UsukiDoll
  • UsukiDoll
I know. What if we let A be the inverse of B?
UsukiDoll
  • UsukiDoll
that would be Matrix B A = B^-1 BB^-1 = identity matrix
Callisto
  • Callisto
What if \(A \ne B^{-1}\) and \(B \ne A^{-1}\) ?
UsukiDoll
  • UsukiDoll
I think that the only way that occurs is if A and B are singular
Callisto
  • Callisto
Hmm.. I'm sorry that I don't know how to do it :( @hartnn Would you mind giving a hand here? The problem we are discussing now is: Let A and B be n x n matrices. Show that if AB is nonsingular then A and B must be nonsingular.(Hint Use Theorem 2.9) Theorem 2.9 states that the homogeneous system of n linear equations in n unknowns Ax = 0 has a nontrivial solution if and only if A is singular.
hartnn
  • hartnn
is it necessary to use that theorem ? can't we go like this way ? if AB is non-sing., \(|AB| \ne 0 \implies |A||B|\ne 0\) for that both |A|and |B| must not =0, so...
hartnn
  • hartnn
if need to use contradiction, then let any1 of the A or B be singular, say A , then |A| =0, then |A||B|=0 , then |AB|=0 which is a contardiction....hence, neither A nor B can be 0
hartnn
  • hartnn
*contradiction, * neither A nor B can be singular
UsukiDoll
  • UsukiDoll
omg how did u get that so fast?
hartnn
  • hartnn
because i knew the property that singularity means determinant = 0.....so something related to determinants should be there... and also, |AB|= |A||B|
UsukiDoll
  • UsukiDoll
I haven't learned determinants yet. That's chapter 3 in my book
UsukiDoll
  • UsukiDoll
What if I'm forced to use that Theorem? then what?
hartnn
  • hartnn
but can we use that if AB is non-singular, then |AB| not =0 ....grrrr, i guess not ..
UsukiDoll
  • UsukiDoll
nnooo...
UsukiDoll
  • UsukiDoll
using advanced things aren't allowed otherwise my prof will get suspicious.
UsukiDoll
  • UsukiDoll
determinants is a topic that my class didn't even learn yet, so if I put that on my proof, then I will get into trouble
UsukiDoll
  • UsukiDoll
he was like DON'T USE ANYTHING THAT WASN'T INTRODUCED!!!!!!!!! IT'S A BAD IDEA
hartnn
  • hartnn
ok ok wait....
hartnn
  • hartnn
whats wrong with this logic ? " Theorem 1.5 that it was AB=BA=In. B is the inverse of A. hmmm..if A is singular, yes the inverse doesn't exist. Let's have B = the inverse of A. Matrix A is singular Inverse doesn't exist so B doesn't exist AB is impossible to achieve unless A is nonsingular "
Callisto
  • Callisto
What if B =/= inverse of A?