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1. Factorizing determinant
Hi, I'm a bit stuck on factorizing the following:
. I can't really see how to start i tried subtracting column 3 from column 1 to give
2. Re: Factorizing determinant
I'd just do it the long way.

With a 2x2 matrix, the determinant of

is ad-bc. For a 3x3 matrix

you need to split it into 3 2x2 determinants
a is associated with
For the rest, look at the rows underneath and omit the column of the letter in top row.
Now you add each of these BUT the sign of them alternate + - + etc.

Wikipedia will explain it better than me about why this is the case.
Last edited by WarriorInAWig; 14-07-2012 at 11:04.
3. Re: Factorizing determinant
The trick with these is normally to make some of the entries 0.

Try subtracting the first row from the second and third row. Does that help you?
4. Re: Factorizing determinant
(Original post by notnek)
The trick with these is normally to make some of the entries 0.

Try subtracting the first row from the second and third row. Does that help you?
hmm: i ended up with . I then tried to take the determinant and got: . Not really too sure if this is the right approach.
5. Re: Factorizing determinant
(Original post by music lover)
hmm: i ended up with . I then tried to take the determinant and got: . Not really too sure if this is the right approach.

You made a sign error.

Once you've made the correction, use difference of two squares to factorise all the quadratics.
Last edited by notnek; 14-07-2012 at 12:32.
6. Re: Factorizing determinant
Off the top of my head; two rows being equal mean the determinant is 0. So (a-b) is a factor of the determinant. There are two other pretty obvious factors, then I think you can just equate coefficients.
7. Re: Factorizing determinant
(Original post by Slumpy)
Off the top of my head; two rows being equal mean the determinant is 0. So (a-b) is a factor of the determinant. There are two other pretty obvious factors, then I think you can just equate coefficients.
Where do you get this from?
8. Re: Factorizing determinant
(Original post by notnek)

You made a sign error.

Once you've made the correction, use difference of two squares to factorise all the quadratics.
hmm ok so now i have:

On a side note is [latex]b^2c=c^2b{/latex]? I'm guessing not I tested it with a few numbers.
9. Re: Factorizing determinant
(Original post by music lover)
Where do you get this from?
If a=b, we get:

This has a determinant of 0, so we know that if a=b, the determinant is 0, thus a-b is a factor of it. This is often a useful trick.
10. Re: Factorizing determinant
(Original post by music lover)
hmm ok so now i have:
You're still making sign errors and not simplifying properly.

If you do the row operation then the left middle entry becomes:

And since 1-1=0 this simplifies to

which can be factorised:

Now do the same with the other entries and then find the determinant. Let me know if there's any part that you're not sure about.

On a side note is ? I'm guessing not I tested it with a few numbers.
No, it is not true in general.
Last edited by notnek; 14-07-2012 at 13:26.
11. Re: Factorizing determinant
(Original post by notnek)
You're still making sign errors and not simplifying properly.

If you do the row operation then the left middle entry becomes:

And since 1-1=0 this simplifies to

which can be factorised:

Now do the same with the other entries and then find the determinant. Let me know if there's any part that you're not sure about.

No, it is not true in general.
done it thanks very much! Now onto this one. . I need to show it's equal to

Is this right so far: which is equal to . Sorry for all the questions!
Last edited by music lover; 14-07-2012 at 14:04.
12. Re: Factorizing determinant
(Original post by music lover)
done it thanks very much! Now onto this one. . I need to show it's equal to

Is this right so far: which is equal to . Sorry for all the questions!
Again, you should be doing some row/column operations to make some of the entries zero before expanding the determinant.

So subtract the first column from the other two columns.
13. Re: Factorizing determinant
(Original post by notnek)
Again, you should be doing some row/column operations to make some of the entries zero before expanding the determinant.

So subtract the first column from the other two columns.
Ah ok so I have gotten to . Taking the determinant of this gives; but how would i then simplify it further? I did difference of two squares to get and .
Last edited by music lover; 14-07-2012 at 15:34.
14. Re: Factorizing determinant
(Original post by music lover)
Ah ok so I have gotten to . Taking the determinant of this gives; but how would i then simplify it further? I did difference of two squares to get and .
The (2,2) entry is wrong. You should have instead of . I think you may have divided instead of subtracted. The (3,2) entry is also wrong for a similar reason.

From here, I always like to change sin, cos and tan to s,c and t so that I concentrate more on the algebra instead of the trig. Using the determinant becomes:

Now using

and , the determinant simplifies to

Can you finish it off?
Last edited by notnek; 14-07-2012 at 15:58.
15. Re: Factorizing determinant
(Original post by notnek)
The (2,2) entry is wrong. You should have instead of . I think you may have divided instead of subtracted. The (3,2) entry is also wrong for a similar reason.

From here, I always like to change sin, cos and tan to s,c and t so that I concentrate more on the algebra instead of the trig. Using the determinant becomes:

Now using

and , the determinant simplifies to

Can you finish it off?

Thanks very much , yeah I divided, silly mistake. What happened to the 1 with the sec^2 identity? Worked it out :P.
Last edited by music lover; 14-07-2012 at 16:47.
16. Re: Factorizing determinant
(Original post by notnek)
The (2,2) entry is wrong. You should have instead of . I think you may have divided instead of subtracted. The (3,2) entry is also wrong for a similar reason.

From here, I always like to change sin, cos and tan to s,c and t so that I concentrate more on the algebra instead of the trig. Using the determinant becomes:

Now using

and , the determinant simplifies to

Can you finish it off?
Thought I had solved it but I get up to the very last part...

. This becomes

Simplifying this last bit i'm not too sure on.
17. Re: Factorizing determinant
(Original post by music lover)
Thought I had solved it but I get up to the very last part...

. This becomes
Notice here that you have a factor of in both terms so you can take it outside the expression to get:

See if you can fill in the blank and simplify.

A general tip in algebra is to always try to factorise fully before expanding.
Last edited by notnek; 14-07-2012 at 17:12.
18. Re: Factorizing determinant
(Original post by notnek)
Notice here that you have a factor of in both terms so you can take it outside the expression to get:

See if you can fill in the blank and simplify.

A general tip in algebra is to always try to factorise fully before expanding.
done it!!!! Thank you so much
19. Re: Factorizing determinant
(Original post by music lover)
Thought I had solved it but I get up to the very last part...

. This becomes

Simplifying this last bit i'm not too sure on.
On a separate note: you should try to be more careful when expanding. There are many errors in this working. You have said:

Every term in your expansion is wrong. E.g. the first term should be instead of

Or maybe I'm missing something? I'm very confused how you arrived at your expansion.

Edit Now I see what you did. You expanded

Last edited by notnek; 14-07-2012 at 17:24.
20. Re: Factorizing determinant
(Original post by notnek)
On a separate note: you should try to be more careful when expanding. There are many errors in this working. You have said:

Every term in your expansion is wrong. E.g. the first term should be instead of

Or maybe I'm missing something? I'm very confused how you arrived at your expansion.
Yeah i realized after. Epic fail...