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 Post subject: A(A - I) = 0Posted: Mon, 12 Mar 2012 03:50:35 UTC
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Guys why is it that for MATRICES A(A - I) = 0 doesn't imply that A = 0 and A - I = 0 but for real number like for example x(x - 3) = 0 this kind of method apply.

A = any 2x2 matrix
I = Identity matrix

Thanks

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 04:02:39 UTC
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Mathinik wrote:
Guys why is it that for MATRICES A(A - I) = 0 doesn't imply that A = 0 and A - I = 0 but for real number like for example x(x - 3) = 0 this kind of method apply.

A = any 2x2 matrix
I = Identity matrix

Thanks

The real numbers form something called an integral domain, which by definition is a place where non-zero things cannot multiply to 0. Matrices are not an integral domain, so it's to be expected that this would happen.

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 04:48:13 UTC
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Sorry but could you please explain a bit more what is integral domain?

Thanks

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 04:48:42 UTC
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Mathinik wrote:
Sorry but could you please explain a bit more what is integral domain?

Thanks

It's just a place where nonzero things cannot multiply to 0.

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 19:02:48 UTC
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I think a counterexample would help to solidify there exists a nonzero matrix A such that , maybe you should give her one Shadow.

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 19:06:46 UTC
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daveyinaz wrote:
I think a counterexample would help to solidify there exists a nonzero matrix A such that , maybe you should give her one Shadow.

I took it the op already knew of one and wanted a more conceptual reason, but you may be right:

works for the 2x2 case.

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 19:27:03 UTC
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Hello, Mathinik!

Quote:
Why is it that for matrices doesn't imply that or
but for real numbers, for example: ., this kind of method applies.

The Zero-product Rule doesn't apply to matrices.

We can have two nonzero matrices whose product is the zero matrix.

Example: .

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 Post subject: Re: A(A - I) = 0Posted: Mon, 12 Mar 2012 19:36:22 UTC
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It should be noted my example is for a matrix which can multiply by itself to get 0, as this is a fine lack of the so-called "zero product rule", if you want one that works for the A(A-I) you need to pick something like or

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