Post by: panicatthelunchbar on January 18, 2011, 08:17:31 pm
5. The system of equations x+y+z+w=4, x+3y+3z=2, x+y+2z−w=6 has infinitely many solutions.
Describe this family of solutions and give the unique solution when w = 6.
Also, how do you find the inverse of a 3x3 matrix with unknown variables on the CAS and by hand.
Thanks everyone! :)
Post by: dude on January 18, 2011, 08:28:24 pm
Post by: panicatthelunchbar on January 18, 2011, 09:00:48 pm
the matrix is:
2, a, -1
3, 4, -(a+1)
10, 8, a-4
Post by: Romperait on January 18, 2011, 09:18:22 pm
Edit: Just for you information, I didn't see one question on a practice exam regarding 3x3 matrices. I would say it's not on the study design, but my memory can be faulty.
Post by: Andiio on January 18, 2011, 09:21:49 pm
It doesn't work when you put that matrix ^-1?
Edit: Just for you information, I didn't see one question on a practice exam regarding 3x3 matrices. I would say it's not on the study design, but my memory can be faulty.
Don't think 3x3 matrices are on the study design, but it's not really that hard to just input a 3x3 matrix into your CAS calc. :P
Post by: brightsky on January 18, 2011, 09:31:15 pm
Post by: Romperait on January 18, 2011, 09:50:56 pm
Yeah, like everyone has said, 3x3 matrices aren't on the study design, and the 'easiest' way to actually find one is very crude. For the question you posted above, it would be pretty easy to solve the system since x is in all three equations, meaning a simple subtraction of two equations would leave you with only two unknowns to work with. Best way to work with simultaneous equations with more than two variables is to use the elimination method in a step-by-step way, eliminating the number of variables gradually.
Most definitely. Finding the inverse of 3x3 matrices by hand can be a real bother. Elimination is far more doable.
Post by: Pixon on January 18, 2011, 09:56:19 pm
(took Mr Near a good 20mins ;) for those who know him )
Post by: Romperait on January 18, 2011, 10:11:00 pm
Let's face it...hardly anybody even knows they are transposing a 2x2 matrix when they find its inverse. Transposing a 3x3...OH BOY...
(took Mr Near a good 20mins ;) for those who know him )
Best class...............................................................
Edit: Wait a second. We go to the same school!?
Post by: Pixon on January 18, 2011, 10:21:18 pm
Let's face it...hardly anybody even knows they are transposing a 2x2 matrix when they find its inverse. Transposing a 3x3...OH BOY...
(took Mr Near a good 20mins ;) for those who know him )
Best class...............................................................
Edit: Wait a second. We go to the same school!?
I thought I was all alone... :)
Post by: TrueTears on January 18, 2011, 10:32:38 pm
Hi Guys, can someone please help me with these questions. They are from the Essential book. The chapter is just a revision chapter but I don't seem to get the idea of using the parameter/special symbol instead of a numerical value. Why do we have to do this for sim. equations with more than two variables?there are often some tricks to find the inverse of a 3x3 matrix, eg if the matrix is diagonal, upper/lower triangular etc, but in most cases, if the matrix doesnt fall into one of those categories it's quite annoying to find the inverse.
5. The system of equations x+y+z+w=4, x+3y+3z=2, x+y+2z−w=6 has infinitely many solutions.
Describe this family of solutions and give the unique solution when w = 6.
Also, how do you find the inverse of a 3x3 matrix with unknown variables on the CAS and by hand.
Thanks everyone! :)
Post by: pHysiX on February 07, 2011, 11:26:40 am
Thank god for Gauss though :smitten: :smitten: :smitten: *hint learn Gauss :D*
If only VCAA would recognise this and be examinable :P
Post by: QuantumJG on February 07, 2011, 01:41:53 pm
Hi Guys, can someone please help me with these questions. They are from the Essential book. The chapter is just a revision chapter but I don't seem to get the idea of using the parameter/special symbol instead of a numerical value. Why do we have to do this for sim. equations with more than two variables?
5. The system of equations x+y+z+w=4, x+3y+3z=2, x+y+2z−w=6 has infinitely many solutions.
Describe this family of solutions and give the unique solution when w = 6.
Also, how do you find the inverse of a 3x3 matrix with unknown variables on the CAS and by hand.
Thanks everyone! :)
You could put the system of equations like this:
~
So I got (x,y,z,w) = (5 - 1.5t,-3 - 1.5t, 2 + 2t, t) (I may be wrong!!!)
As for find the inverse of a 3x3 matrix. If the det =/= 0 then you can find the characteristic polynomial
e.g.
A =
set |A - λI| = 0
So (1 - λ)3 = 0
So
X3 - 3X2 + 3X - 1 = 0
So
A3 - 3A2 + 3A = I
So
A-1 = 3I - 3A + A2
Otherwise you can use this row reduction method that takes ages!!!!
Post by: cltf on February 07, 2011, 07:23:44 pm
Post by: tram on February 07, 2011, 08:24:16 pm
Post by: panicatthelunchbar on February 07, 2011, 10:08:11 pm
Post by: cltf on February 08, 2011, 06:48:18 pm
Hi Guys, can someone please help me with these questions. They are from the Essential book. The chapter is just a revision chapter but I don't seem to get the idea of using the parameter/special symbol instead of a numerical value. Why do we have to do this for sim. equations with more than two variables?
5. The system of equations x+y+z+w=4, x+3y+3z=2, x+y+2z−w=6 has infinitely many solutions.
Describe this family of solutions and give the unique solution when w = 6.
Also, how do you find the inverse of a 3x3 matrix with unknown variables on the CAS and by hand.
Thanks everyone! :)
You could put the system of equations like this:
~
So I got (x,y,z,w) = (5 - 1.5t,-3 - 1.5t, 2 + 2t, t) (I may be wrong!!!)
As for find the inverse of a 3x3 matrix. If the det =/= 0 then you can find the characteristic polynomial
e.g.
A =
set |A - λI| = 0
So (1 - λ)3 = 0
So
X3 - 3X2 + 3X - 1 = 0
So
A3 - 3A2 + 3A = I
So
A-1 = 3I - 3A + A2
Otherwise you can use this row reduction method that takes ages!!!!
So as it turns out, I asked my teacher, since w=6 sub it in. And then everything just falls into place.