Re: Still ,Matrices, A is square: If AB = Identity, is B the inverse?


"Timothy Murphy" <tim@xxxxxxxxxxxxxxxxxxxxxx> a écrit dans le message de
news: BQXLf.6471$j7.223578@xxxxxxxxxxxxxxxxx
linux wrote:

If In is a matrix the coefficients of which belong to a some ring, the
result is it really still? I think that yes. Can one prove him(it)
without
determinant?

AB = I => BA = I.

One way to see this is to consider the minimal equation of A
(ie the polynomial of minimal degree satisfied by A), say
x^d + a_1x^{d-1} + ... + a_d = 0.
Then a_d <> 0, since otherwise you would get an equation of lower degree
on multiplying on the right by B.
Now it follows that you get a right and left inverse C, say,
as a polynomial in A.
It is easy to show that the inverse is unique, ie B = C.

--
Timothy Murphy

Thanks.
The ring is commutative ?


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