A(A - I) = 0

Mathinik · **Joined:** Mon, 12 Mar 2012 03:46:37 UTC **Posts:** 2

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

Shadow · **Posted:** Mon, 12 Mar 2012 04:02:39 UTC

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.

Mathinik · **Joined:** Mon, 12 Mar 2012 03:46:37 UTC **Posts:** 2

Sorry but could you please explain a bit more what is integral domain?

Thanks

Shadow · **Posted:** Mon, 12 Mar 2012 04:48:42 UTC

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.

daveyinaz · **Joined:** Sat, 16 Aug 2008 04:47:19 UTC **Posts:** 208

I think a counterexample would help to solidify there exists a nonzero matrix A such that A(A-I) = 0

, maybe you should give her one Shadow.

Shadow · **Posted:** Mon, 12 Mar 2012 19:06:46 UTC

daveyinaz wrote:

I think a counterexample would help to solidify there exists a nonzero matrix A such that A(A-I) = 0

, 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:

$\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}$ works for the 2x2 case.

Soroban · **Posted:** Mon, 12 Mar 2012 19:27:03 UTC

Hello, Mathinik!

Quote:

Why is it that for matrices A(A - I) = 0

doesn't imply that A = 0

or

but for real numbers, for example: . x(x - 3) = 0

, 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: . $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix}1 & 1 \\ \text{-}1 & \text{-}1 \end{pmatrix} \;=\;\begin{pmatrix}0 & 0 \\ 0&0 \end{pmatrix}$

Shadow · **Posted:** Mon, 12 Mar 2012 19:36:22 UTC

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 $\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}$ or $\begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}$

S.O.S. Mathematics CyberBoard

A(A - I) = 0

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