Suppose A, B and X are invertible matrices such that BinverseXA = AB:
Find an expression for X in terms of A and B.

- anonymous

- katieb

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- anonymous

\[BX^{-1}A=AB\]

- anonymous

I have a question to see if I'm doing this right. Do I first times each side by the inverse of B to get rid of it then by X??

- anonymous

What does it mean that it is invertible

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## More answers

- anonymous

determinant is not equal to zero

- anonymous

Does that mean I can move them around any special way? I'm so confused

- anonymous

say wuttt

- anonymous

I still don't know how to do this :(

- anonymous

is that from linear algebra class in college?

- anonymous

yes

- anonymous

If the matrix A is invertible, that means there exists a matrix\[A^{-1}\]such that:\[AA^{-1}=A^{-1}A=I\]where I is the identity matrix.

- anonymous

Okay I know that but I still want to know if what I wrote up there is on the right track

- TuringTest

yes, multiply both sides by \(B^{-1}\) and what do you get?

- anonymous

you get \[x^{-1} A = B^{-1}AB\]
so I went on and multiplied each side by X and then got
\[A=XB^{-1}AB\]
But I'm confused beacuse it doesn't look right

- TuringTest

I'm having a little trouble too actually :/
while I work on it, I can say that you can use \(AB=B^{-1}AB\) to make that a little prettier, but I still can't solve for \(X\) yet

- anonymous

sorry my laptop decided to restart lol. So you're not allowed to rearrange the letters at all, right?

- anonymous

or where did you get the B to? on the right side

- TuringTest

no, matrix multiplication is not commutative (cant move the letters)

- TuringTest

oh I see, I misread the question, sorry

- TuringTest

\[BX^{-1}A=AB\] all matrices are invertible
solve for \(X\)
correct?

- anonymous

Yup!

- anonymous

I tried switching them around in different ways but nothing works lol

- TuringTest

yeah, there must be some trick we're missing :/

- anonymous

Like there's no way to rearrange it so that BA=AB ?

- TuringTest

only if BA and AB are inverses of each other, which I am trying to prove... so far unsuccessfully.

- TuringTest

I mean if A and B are inverses of each other, then AB=BA=I

- anonymous

I think I'll just leave it and wait for the solution to be posted online to understand it lol thanks a lot for your help though!

- TuringTest

This will continue to bother me, please let me know the answer when you find out :)
Sorry I couldn't really help.

- anonymous

I really appreciate that you tried! I'll post the answer when it's up!

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