Question

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

Answer 1

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

Answer 2

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??

Answer 3

What does it mean that it is invertible

Answer 4

determinant is not equal to zero

Answer 5

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

Answer 6

say wuttt

Answer 7

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

Answer 8

is that from linear algebra class in college?

Answer 9

yes

Answer 10

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.

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

oh I see, I misread the question, sorry

Answer 19

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

Answer 20

Yup!

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

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!

Answer 27

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

Answer 28

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