Re: Still ,Matrices, A is square: If AB = Identity, is B the inverse?
- From: Timothy Murphy <tim@xxxxxxxxxxxxxxxxxxxxxx>
- Date: Sat, 25 Feb 2006 12:12:24 +0000
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
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.
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