Abstract Algebra Matrix Formulas

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September 6th 2012, 12:49 AM
TheHowlingLung

Abstract Algebra Matrix Formulas

Find a formula for the n-th power of any 2 x 2 upper-triangular real matrix,
a...b
0...d
where n is a positive integer, and verify the formula by induction on n. Then, assuming a and d are nonzero, show the matrix is invertible and determine if there is a single formula that works for all integers n (i.e., when n < 0 too).

So, first in order to derive a general formula for the matrix, I did out the multiplication. The second term in the first row is causing me problems. I find that the first term in the first row is a^n, and the last term in the last row is d^n, but the second term in the first row is causing problems. Below are my results for the second term for each n:
n=1 : b
n=2: ab + bd
n=3: a^2b + abd + bd^2
n=4: a^3b + a^2bd + abd^2 + bd^3

I see a pattern, but am not sure how to represent the pattern in terms of n, so any help with this would be appreciated. I was thinking of something along the lines of a^(n-1)b + a^(n-2)bd... but I'm not really sure that's the way to go. Also, any other tips/advice while working on the rest of the problem would be awesome, and if there is a formula that works for negative integers of n as well.
September 6th 2012, 01:24 AM
a tutor

Re: Abstract Algebra Matrix Formulas

Take the b out and things look a little clearer.

$b$
$b(a+d)$
$b(a^2+ad+d^2)$
$b(a^3+a^2d+ad^2+d^3)$

The expression in brackets can be written neatly as sum.
September 6th 2012, 01:49 AM
emakarov

Re: Abstract Algebra Matrix Formulas

Also, $\sum_{k=0}^na^{n-k}d^k=(a^{n+1}-d^{n+1})/(a-d)$ .
September 6th 2012, 02:54 AM
TheHowlingLung

Re: Abstract Algebra Matrix Formulas

Quote:

Originally Posted by emakarov

Also, $\sum_{k=0}^na^{n-k}d^k=(a^{n+1}-d^{n+1})/(a-d)$ .

I don't fully understand this. My experience with sum notation is brief. Could you explain this a little? When I look at that sum, I see it as a^n-0d⁰+a^n-1d¹...+ a^n-ndⁿ. Is that incorrect? Is that the correct way of representing the summation that's happening as n increases? I don't understand where you are getting (aⁿ⁺¹-dⁿ⁺¹)/(a-d). Maybe I'm missing something?
September 6th 2012, 03:04 AM
a tutor

Re: Abstract Algebra Matrix Formulas

Example

$a^3-d^3=(a-d)(a^2+ad+d^2)$

$\frac{a^3-d^3}{a-d}=a^2+ad~+d^2$
September 6th 2012, 03:09 AM
emakarov

Re: Abstract Algebra Matrix Formulas

Quote:

Originally Posted by TheHowlingLung

When I look at that sum, I see it as a^n-0d⁰+a^n-1d¹...+ a^n-ndⁿ.

This is correct.

Quote:

Originally Posted by TheHowlingLung

I don't understand where you are getting (aⁿ⁺¹-dⁿ⁺¹)/(a-d).

The easiest way to prove this is to multiply (a - d) by (aⁿd⁰+a^n-1d¹...+ a⁰dⁿ); most terms cancel out. See also here and here.
September 6th 2012, 03:13 AM
TheHowlingLung

Re: Abstract Algebra Matrix Formulas

Thanks guys! I appreciate the help.
September 6th 2012, 03:14 AM
emakarov

Re: Abstract Algebra Matrix Formulas

Quote:

Originally Posted by emakarov

This is correct.

The easiest way to prove this is to multiply (a - d) by (aⁿd⁰+a^n-1d¹...+ a⁰dⁿ); most terms cancel out. See also here and here.

Edit: The first link does not work, but it's in the Math Help Boards forum. The part of the URL after the site name is correct.