## KatClaire 2 years ago Suppose A, B and X are invertible matrices such that BinverseXA = AB: Find an expression for X in terms of A and B.

1. KatClaire

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

2. KatClaire

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??

3. KatClaire

What does it mean that it is invertible

4. modphysnoob

determinant is not equal to zero

5. KatClaire

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

6. modphysnoob

say wuttt

7. KatClaire

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

8. modphysnoob

is that from linear algebra class in college?

9. KatClaire

yes

10. joemath314159

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.

11. KatClaire

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

12. TuringTest

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

13. KatClaire

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

14. 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

15. KatClaire

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

16. KatClaire

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

17. TuringTest

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

18. TuringTest

oh I see, I misread the question, sorry

19. TuringTest

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

20. KatClaire

Yup!

21. KatClaire

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

22. TuringTest

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

23. KatClaire

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

24. TuringTest

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

25. TuringTest

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

26. KatClaire

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!

27. TuringTest

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

28. KatClaire

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