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#1 2011-02-22 13:44:45
- Fifa
- Member
- Registered: 2010-01-15
- Posts: 6
Matrix algebra: Prove AB-BA cannot equal I2?
The question for my matrix algebra class is: show that there is no 2x2 matrix A and B such that AB-BA= I2 (I sub 2, identity matrix, sorry can't write I sub2)
I2: |1 0|
|0 1|
We obviously can't prove it with a specific example, it has to work for all 2x2 matrices. anyone know how to prove this? Thanks for the help!
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#2 2011-02-22 14:04:58
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Matrix algebra: Prove AB-BA cannot equal I2?
Hi fifa;
Let us say it is true and fight our way to a contradiction.
Form two generic matrices A and B
If the statement is true then this is true for some a-h.
Do the multiplication and subtraction.
What this means is that
Then:
Also
Substituting 2) into 3) we get:
We have a contradiction therefore no such 2x2 matrices exist.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#3 2011-02-22 14:15:12
- Fifa
- Member
- Registered: 2010-01-15
- Posts: 6
Re: Matrix algebra: Prove AB-BA cannot equal I2?
I understand everything you did, but I am still confused on how bg-cf=1 can disprove this.
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#4 2011-02-22 14:16:22
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Matrix algebra: Prove AB-BA cannot equal I2?
Please check the post again for the explanation. Do you see it now?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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#5 2011-02-22 14:25:04
- Fifa
- Member
- Registered: 2010-01-15
- Posts: 6
Re: Matrix algebra: Prove AB-BA cannot equal I2?
Oh haha the rest of it wasn't up yet, now it makes sense, thanks so much!
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#6 2011-02-22 14:27:26
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Matrix algebra: Prove AB-BA cannot equal I2?
Your welcome! Hope it is what you want.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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