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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: luken93 on March 23, 2010, 09:06:24 pm

Title: help with 3x3 matrix
Post by: luken93 on March 23, 2010, 09:06:24 pm: Firstly someone needs to explain to me how to do matrix on latex
Secondly,

If A= $[2,0,0]$
$[0,0,2]$
$[0,0,0]$

Find $A^2$ and hence find $A^-1$

I've found $A^2$ but having a bit of trouble with the inverse

Thanks
Title: Re: help with 3x3 matrix
Post by: m@tty on March 23, 2010, 09:12:21 pm: $A= \begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&0&0\end{bmatrix}$

$A^2=\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&0&0\end{bmatrix}\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&0&0\end{bmatrix}=\begin{bmatrix} 4&0&0 \\ 0&0&0 \\0&0&0\end{bmatrix}$

I didn't do CAS, so I don't know what you know. But any matrix that has a row consisting of only zeros, has a determinant of zero, and hence has no inverse.

Also, to show a matrix:

Code: [Select]
\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&0&0\end{bmatrix}
That is, begin with \begin{bmatrix} then enter each element of the first row, separated by '&', when you get to the end of the row insert '\\' and repeat completing each row until you finish the matrix. When you get to the very end place \end{bmatrix}. :)
Title: Re: help with 3x3 matrix
Post by: luken93 on March 23, 2010, 10:34:06 pm: ok thanks, just realised i made an error

A= $\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&2&0\end{bmatrix}$ , not

A= $\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&0&0\end{bmatrix}$

now can you redo the question haha
Title: Re: help with 3x3 matrix
Post by: m@tty on March 23, 2010, 10:39:58 pm: Well thanks :buck2:

$A= \begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&2&0\end{bmatrix}$

$A^2=\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&2&0\end{bmatrix}\begin{bmatrix} 2&0&0 \\ 0&0&2 \\0&2&0\end{bmatrix}=\begin{bmatrix} 4&0&0 \\ 0&0&4 \\0&4&0\end{bmatrix}$

I didn't do CAS, so I don't know what you know. ~~But any matrix that has a row consisting of only zeros, has a determinant of zero, and hence has no inverse.~~ <-Irrelevant now.

Well yeah, I still don't know what you know. What have you been told about determinants/square matrices/power matrices?..
Title: Re: help with 3x3 matrix
Post by: luken93 on March 23, 2010, 10:55:05 pm: aah not much, i know how to get inverse of 2x2 but im not really sure how to do it for a 3x3

i looked in the solutions pdf from here and they show that you set up

$a^2*a^-1=a$

hence

$A^-1= \begin{bmatrix} a&b&c \\ d&e&f \\g&h&i\end{bmatrix}$

therefore $\begin{bmatrix} 4&0&0 \\ 0&0&4 \\0&4&0\end{bmatrix} \begin{bmatrix} a&b&c \\ d&e&f \\g&h&i\end{bmatrix} = A$

but somehow they get that a=2, whereas i thought it was 4?
Title: Re: help with 3x3 matrix
Post by: m@tty on March 23, 2010, 11:09:35 pm: Ok so:

$A=A^2A^{-1}=\begin{bmatrix} a&b&c \\ d&e&f \\g&h&i\end{bmatrix}\begin{bmatrix} 4&0&0 \\ 0&0&4 \\0&4&0\end{bmatrix}=\begin{bmatrix} 4a&4c&4b \\ 4d&4f&4e \\4g&4i&4h\end{bmatrix}=\begin{bmatrix}2&0&0 \\0&0&2\\0&2&0\end{bmatrix}$

$\therefore a=\frac{1}{2};e=\frac{1}{2};i=\frac{1}{2}$ and all other elements are equal to zero.

$\therefore A^{-1}=\begin{bmatrix}\frac{1}{2}&0&0\\0&\frac{1}{2}&0\\0&0&\frac{1}{2}\end{bmatrix}$

$=\frac{1}{2}I_3$ where $I_3$ is the 3x3 identity matrix.

Is this how you are meant to deduce it?
Title: Re: help with 3x3 matrix
Post by: luken93 on March 23, 2010, 11:22:42 pm: aah ok i get it now, just wasn't looking at the 4a 4c 4b etc = a properly, i was thinking that it shudve been 4, 0, 0

couldve solved it ages ago hahaha
Title: Re: help with 3x3 matrix
Post by: luken93 on March 23, 2010, 11:23:20 pm: next test: UniMelb Maths Comp
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 05:43:20 pm: i'm pretty sure that 3x3 matricies are not part of the study design or mm cas, only 2x2. They're just there if you're interested. You don't even reli have and pure matricies questions anyway-i've never come across one. You only use matricies for probablility.

And even if 3x3 matricies were on the course, i'm postive you would've had to find inverses, the only way of doing that is with uni maths techniques

However matracies are soooo cool once you learn more about them. Uni maths is awesome for taking all this kinda stuff further and making maths interesting. Pretty sure m@tty would agree with me.
Title: Re: help with 3x3 matrix
Post by: kyzoo on March 28, 2010, 07:04:27 pm: Quote from: tram on March 28, 2010, 05:43:20 pm
And even if 3x3 matricies were on the course, i'm postive you would've had to find inverses, the only way of doing that is with uni maths techniques

However matracies are soooo cool once you learn more about them. Uni maths is awesome for taking all this kinda stuff further and making maths interesting. Pretty sure m@tty would agree with me.

Uni Maths Matrices is the most annoying thing >.<. If you screw up the arithmetic just once, then you have to start over again. Finding the inverse of a 3x3 matrix take likes...30-40 arithmetic operations =/
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 08:12:45 pm: ^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed
Title: Re: help with 3x3 matrix
Post by: luken93 on March 28, 2010, 08:16:26 pm: it was only 1 question in matrices chapter extended response, just wanted to know how to do it just incase our teacher sprung it on us.
Title: Re: help with 3x3 matrix
Post by: kyzoo on March 28, 2010, 08:50:57 pm: Quote from: tram on March 28, 2010, 08:12:45 pm
^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed

Too bad we aren't allowed to use calculators for final exam =(
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 09:01:03 pm: You aren't allowed calculators? Not even a scientific?
Title: Re: help with 3x3 matrix
Post by: kyzoo on March 28, 2010, 09:05:07 pm: Directly quoted from the book

Quote
You will NOT be allowed to bring any of the following into your 620-190 examinations in November, nor will you be provided with them

~ A calculator (of any kind)
~ Any form of notes or summary ('cheat') sheet
~ A formula sheet
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 09:06:02 pm: Quote from: kyzoo on March 28, 2010, 08:50:57 pm
Quote from: tram on March 28, 2010, 08:12:45 pm
^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed

Too bad we aren't allowed to use calculators for final exam =(

yup

btw luken, you should probs ask someone who know more about the mm course than me to verify that 3x3 matricies aren't in the study design. I'm about 90% sure they're not

Quote from: m@tty on March 28, 2010, 09:01:03 pm
You aren't allowed calculators? Not even a scientific?

nope, ALL by hand, we are doing-as my head of maths once said- " propper maths, not that vce rubbish"
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 09:08:58 pm: You can bring NOTHING in with you :o

Do you have only one exam, in November? Such a lot to fit into your brain.

We can bring a calculator and as much notes as we like (I think) into each of our two exams...
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 09:24:57 pm: woah, should've chosen monash >.<

Ah well, Umm, i'm pretty sure, and kyzoo can confirm this, we have two exams one at the start of term three and one after the normal vce exams at the end of the year.
Title: Re: help with 3x3 matrix
Post by: kyzoo on March 28, 2010, 09:50:39 pm: Quote from: m@tty on March 28, 2010, 09:08:58 pm
You can bring NOTHING in with you :o

Do you have only one exam, in November? Such a lot to fit into your brain.

I'm fine with it haha, after not having a bound book at all for Methods

Quote from: tram on March 28, 2010, 09:24:57 pm
woah, should've chosen monash >.<

Ah well, Umm, i'm pretty sure, and kyzoo can confirm this, we have two exams one at the start of term three and one after the normal vce exams at the end of the year.

Err, I think we only have the one after VCE exams

Edit: Actually it's two November exams
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 09:55:30 pm: Methods vs. UMEP hmmmm....

I think UMEP has infinitely more knowledge and thinking required.

EDIT:
With Methods you don't really need a bound reference, there is almost no use for them at all.
Probably going to be tending towards the same situation by the end of the year with Uni Maths, assuming there is nothing incredibly hard.
Title: Re: help with 3x3 matrix
Post by: superflya on March 28, 2010, 10:17:15 pm: Quote from: tram on March 28, 2010, 09:06:02 pm
Quote from: kyzoo on March 28, 2010, 08:50:57 pm
Quote from: tram on March 28, 2010, 08:12:45 pm
^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed

Too bad we aren't allowed to use calculators for final exam =(

yup

btw luken, you should probs ask someone who know more about the mm course than me to verify that 3x3 matricies aren't in the study design. I'm about 90% sure they're not

Quote from: m@tty on March 28, 2010, 09:01:03 pm
You aren't allowed calculators? Not even a scientific?

nope, ALL by hand, we are doing-as my head of maths once said- " propper maths, not that vce rubbish"

there are 3x3 matrices in the methods course, but ur not expected to know how to find the inverse and such without a calculator.
Title: Re: help with 3x3 matrix
Post by: TrueTears on March 28, 2010, 10:41:09 pm: Quote from: tram on March 28, 2010, 08:12:45 pm
^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed
u don't need 15 steps to work out a 3x3 matrix, that's the long method... in fact for any n x n matrix to find the inverse is quite easy...
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 10:43:08 pm: How would you do it TT?
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 10:44:44 pm: Quote from: kyzoo on March 28, 2010, 09:50:39 pm
Quote from: m@tty on March 28, 2010, 09:08:58 pm
You can bring NOTHING in with you :o

Do you have only one exam, in November? Such a lot to fit into your brain.

I'm fine with it haha, after not having a bound book at all for Methods

Quote from: tram on March 28, 2010, 09:24:57 pm
woah, should've chosen monash >.<

Ah well, Umm, i'm pretty sure, and kyzoo can confirm this, we have two exams one at the start of term three and one after the normal vce exams at the end of the year.

Err, I think we only have the one after VCE exams

Edit: Actually it's two November exams

I'm looking at the timetable Ms Michelle gave us and it says that we have a "mid year test" 1st lecture back in term three on top of the final single exam. Not sure if this counts as an exam tho.
Quote from: TrueTears on March 28, 2010, 10:41:09 pm
Quote from: tram on March 28, 2010, 08:12:45 pm
^definitely, lol, caculators make it sooooo much easier. But if you do it the right way, you usually only need like....~15 steps. But yea....defs, one TINY mistake at the start means you're screwed
u don't need 15 steps to work out a 3x3 matrix, that's the long method... in fact for any n x n matrix to find the inverse is quite easy...

cofactor method rite?
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 10:46:09 pm: shit, that's only to find the det, How do you find inverses quickly tt?
Title: Re: help with 3x3 matrix
Post by: tram on March 28, 2010, 10:47:30 pm: and does the easy way work for m x n matricies or just square matricies?
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 10:54:45 pm: Only square matrices have inverses....
Title: Re: help with 3x3 matrix
Post by: TrueTears on March 28, 2010, 11:05:08 pm: Linear algebra by david lay, pg 114 - read example explanation in there, cbf typing matrices up on latex

or read the entire book that is your guide to linear algebra success
Title: Re: help with 3x3 matrix
Post by: m@tty on March 28, 2010, 11:18:08 pm: Ahh, can you outline the method? I don't have that book...
Title: Re: help with 3x3 matrix
Post by: TrueTears on March 28, 2010, 11:26:55 pm: http://www.megaupload.com/?d=BD8D7E68

now u have it.

make sure if u have time, read that entire book, it's probs one of the best books out there i've read for lin alg
Title: Re: help with 3x3 matrix
Post by: Mao on March 30, 2010, 11:46:05 am: These two posts contain mistakes.
http://vcenotes.com/forum/index.php/topic,24378.msg248026.html#msg248026 - $A^2 = 4 I_3$

http://vcenotes.com/forum/index.php/topic,24378.msg248044.html#msg248044, since A^2 is incorrect, the final answer is incorrect.

The proper working should be:

$\begin{align*} A^2 &= 4I_3 \\ \implies A^2 A^{-1} & = 4 I_3 A^{-1}\\ A & = 4 A^{-1} \\ A^{-1} & = \frac{1}{4}A\\ \end{align*}$
Title: Re: help with 3x3 matrix
Post by: m@tty on March 30, 2010, 11:55:48 am: Indeed they do :|...

Thank you for rectifying such stupid errors.