Math Is Fun Forum

mikau

Member

Registered: 2005-08-22

Posts: 1,504

matrices

reviewing some linear algebra.

Is it true that for an invertible nxn matrix A, that

that is, if A has an inverse, does the inverse works from either side? Don't tell me how to prove it, i just want to know if its true.

Last edited by mikau (2008-07-24 16:47:20)

---

A logarithm is just a misspelled algorithm.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: matrices

Yes.

http://en.wikipedia.org/wiki/Inverse_matrix

---

Wrap it in bacon

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: matrices

If an element has a right inverse and a left inverse, then both left and right inverses must be equal. This is true in any semigroup with an identity element. (A semigroup is a nonempty set S together with a binary operation that is associative in S.)

Proof: If

, then

It also follows that this inverse is unique.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: matrices

thanks jane, but didn't I tell you not to prove it?
fortunately,  I got that much of the proof, but that assumes initially that, given ax = 1, there is some matrix y such that ya = 1 (has a left/right inverse implies has a right/left inverse). If we can show one exists, then we can show by the above proof that they must be equal. 

I'm simply trying to prove that, if Ax = 1 implies  xA = 1.

for one, I noted Ax = 1  implies (Ax)A = A(xA) = A, but I'm not sure if AM = A implies M = I, and so the problem reduced to trying to prove that.

---

A logarithm is just a misspelled algorithm.