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.

Re: Abstract Algebra Matrix Formulas

Take the b out and things look a little clearer.

$\displaystyle b$

$\displaystyle b(a+d)$

$\displaystyle b(a^2+ad+d^2)$

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

The expression in brackets can be written neatly as sum.

Re: Abstract Algebra Matrix Formulas

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

Re: Abstract Algebra Matrix Formulas

Quote:

Originally Posted by

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

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-0}d^{0}+a^{n-1}d^{1}...+ a^{n-n}d^{n}. 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 (a^{n+1}-d^{n+1})/(a-d). Maybe I'm missing something?

Re: Abstract Algebra Matrix Formulas

Example

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

$\displaystyle \frac{a^3-d^3}{a-d}=a^2+ad~+d^2$

Re: Abstract Algebra Matrix Formulas

Quote:

Originally Posted by

**TheHowlingLung** When I look at that sum, I see it as a^{n-0}d^{0}+a^{n-1}d^{1}...+ a^{n-n}d^{n}.

This is correct.

Quote:

Originally Posted by

**TheHowlingLung** I don't understand where you are getting (a^{n+1}-d^{n+1})/(a-d).

The easiest way to prove this is to multiply (a - d) by (a^{n}d^{0}+a^{n-1}d^{1}...+ a^{0}d^{n}); most terms cancel out. See also here and here.

Re: Abstract Algebra Matrix Formulas

Thanks guys! I appreciate the help.

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 (a

^{n}d

^{0}+a

^{n-1}d

^{1}...+ a

^{0}d

^{n}); 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.