Proof Adj characteristic

**matrix3599** · Mar 19th 2010, 09:14 AM

Hi,

How can I proof that adj(AB)=adj(B)*adj(A)?

I tried to use with the characteristic of A*adj(A)=|A|*I

Thanks.

**tonio** · Mar 19th 2010, 04:32 PM

Originally Posted by matrix3599

Hi,

How can I proof that adj(AB)=adj(B)*adj(A)?

I tried to use with the characteristic of A*adj(A)=|A|*I

Thanks.

If A,B are invertible matrices the proof is very easy and short, but in the general case it is, imo, a very involved, long proof which uses, if I remember correctly, exterior products and devilish stuff like that. Even worse, the only proof I have is veryyyyy long...and in spanish.

Tonio

**Opalg** · Mar 20th 2010, 01:01 AM

Originally Posted by tonio

If A,B are invertible matrices the proof is very easy and short, but in the general case it is, imo, a very involved, long proof which uses, if I remember correctly, exterior products and devilish stuff like that. Even worse, the only proof I have is veryyyyy long...and in spanish.

As an analyst, I would deduce the general case from the invertible one by adding $\varepsilon I$ to A and B (making them invertible), and then letting $\varepsilon\to0$ . But of course that argument won't work over the sort of fields that algebraists like to use.

**tonio** · Mar 20th 2010, 04:40 AM

Originally Posted by Opalg

As an analyst, I would deduce the general case from the invertible one by adding $\varepsilon I$ to A and B (making them invertible), and then letting $\varepsilon\to0$ . But of course that argument won't work over the sort of fields that algebraists like to use.

I suppose that's easier than the algebraic stuff, but then we'd need quite a few things to clear out:

1) Is it true that if $A$ is singular then $\exists\, \epsilon\in\mathbb{R}\,\,\,s.t.\,\,\,\epsilon I+A$ is invertible? Of course it is since the set of singular nxn matrices is an open set in the set of all nxn matrices,

with the euclidean topology inherited from $\mathbb{R}^{n^2}$ , but this is a little work the OP would have to do.

2) Thus, assuming the case for invertible matrices, we'd have that

$adj\left((\epsilon I+A)(\epsilon I+B)\right)=adj(\epsilon I+B)adj(\epsilon I+A)$ (we can take an $\epsilon$ that'll work for both matrices), and then another job for the OP would be to

prove that $adj(\epsilon I+A)\xrightarrow [\epsilon\to 0]{} adj A$ , which can prove to be a little messy since the entries of $adj A$ are (n-1)x(n-1) determinants (the minors of A), so it'd probably

have to rely on the limit of functions composition (though this should not be a big problem, I think, since the determinant function is continuous on the set of all

square matrices)...

So it still looks like a not-too-short way to prove it...but still it looks easier than the algebraic one.

Tonio

**matrix3599** · Mar 20th 2010, 11:42 AM

ops, I forgot to said that it's invertible matix.

tonio, thak you.
easy proof , but I am looking for the Linear proof.

**tonio** · Mar 20th 2010, 12:52 PM

Originally Posted by matrix3599

ops, I forgot to said that it's invertible matix.

tonio, thak you.
easy proof , but I am looking for the Linear proof.

Well, as already said, the case for both matrices invertible is easy:

$adj(A)=A^{-1}\,|A|\cdot I\,,\,adj(B)=B^{-1}\,|B|\cdot I\Longrightarrow \,adj(AB)=(AB)^{-1}$ $|AB|\cdot I=B^{-1}A^{-1}\,|A||B|\cdot I$ ...and now use that scalar

matrices commute with ANY matrix and we're done.

Tonio

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