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**princess snowwhite****Member**- Registered: 2012-11-06
- Posts: 29

let P be any n*n square matrix whose row sum equals 1 then for any postive m the row sum of the matrix P^m equals 1 ,state true or false

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,255

Hi princess snowwhite;

I am not sure about the term row sum. If you mean that each row sums to 1 then the matrix is called stochastic and is used in Markov processes. Every integer power of that matrix will also be stochastic meaning the rows sum to 1. So, I would say yes.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**princess snowwhite****Member**- Registered: 2012-11-06
- Posts: 29

Yes It Does Mean That Each Roq Sums To 1

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,255

Then I would say the answer is yes but I do not remember how to prove that.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi princess snowwhite

Just proved it is true for all 2 x 2 matrices (algebraically). Don't think I can generalise that method of proof though.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Just gone a step further, which may then lead to an inductive proof.

If A and B are 2 x 2 matrices having the property, the AB has the property.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

I think I can put together a proof for n x n if you wish.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Hi Bob

How would you do an iinductive proof here?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

Good question. I abandoned that idea and went for a aij type proof instead. It's just in my head so who knows how it will look when it goes down in type.

Bob

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