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--- Quote from: zzdfa on July 15, 2010, 08:02:39 pm ---Another way you could prove that is to write and then use the binomial theorem to expand that.


--- End quote ---

err could you explain how that works? I can't see how that explains it

zzdfa:
Well after typing it up I see that its not as quick as I thought it was, I thought the last step (*) would be more straightforward. I'll post it anyway since the technique is occaisonally useful.

We want to show that A^n is never equal to the identity.
which is the same as showing that (I+B)^n is never the identity , where

By the binomial theorem, so that if (I+B)^n = I then





(*) and so the problem reduces to showing that the expression is never 0. I can't think of any quick way to do this for this particular B (apart from induction, which would make it too similar to your proof). Oh well, this would work with a more convenient b, (say the matrix with a  single 1 in the top left).

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Ah, thanks zzdfa, that certainly looks useful

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In the video lectures, the guy takes the determinant of the matrix:



By multiplying together the determinants of

and

Is there a general rule for this sort of stuff? I don't remember seeing it before

(I know how to take the determinant by interchanging the top two rows and multiplying along the diagonal, but is he using a different principle?)



Also, how would you go about proving that for is the ONLY centre of ?

i.e. for all square matrices of dimension .

Thanks

zzdfa:
For your first question, you probably saw the example and probably realized that 'the det of a block diagonal matrix is the product of the det of the blocks'. This is true and easy to prove (using Leibniz's formula).

The natural generalization is 'the det of a block matrix is the det of the matrix of the det of the blocks'. (see how the diagonal case is a special case of this because the det ofa  diagonal matrix is the product of its entries). Unfortunately this is not true. See
http://en.wikipedia.org/wiki/Determinant#Block_matrices

And you were asking about 'general rules', another 'general rule' you could use is that the matrix you gave above is a permutation matrix (this particular matrix swaps entries 1 and 2 when you let it act on a vector) and the det of a permutation matrix is just the sign of the permutation, which is -1 in this case since it swaps 2 elements.

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