Solving Equation ABXC = D with Matrix Inverses

[twoflower]

Hi all,

I have this equation:

ABXC = D

Where A, B and C are regular matrixes. The task is to express the matrix X using matrixes A^-1, B^-1, C^-1, D, where A^-1 means inverse matrix.

I don't have any idea how to solve it..

Thank you for any help.

 

[arildno]

Welcome to PF!
1) Now, you know ONE property about the inverse of a matrix, for example:
C*C^-1=C^-1*C=I
where I is the identity matrix.
2) You also know that for any matrix W and identity matrix I, you have:
WI=IW=W
3) You should also know that the product of matrices is ASSOCIATIVE, that is for matrices A, B, C, we have:
A*B*C=(A*B)*C=A*(B*C)

Use these properties.

 

[twoflower]

Thank you, I already tried to use these properties, but without success. I always end with the fact that I cannot simply move X to the right side of the equation, in order to get something like this:
X = D / ABC, because dividing of matrix is not defined. I just need some hint. Unfortunately I'm not able to solve it using just the properties so far...

 

[arildno]

I'll give you a start:
1)Define the matrix W=ABX
2) Hence, your equation can be written as:
WC=D
3) NOW, Apply C^-1 to this equation:
WC*C^-1=DC^-1
4) On your left-hand side, you may now simplify:
W=DC^-1
5) Or, expressed with your original matrices:
ABX=DC^-1
6) Can you now try to proceed further along these lines?

 

[twoflower]

Thank you arildno, that's exactly I was asking for - this hint (multiplicating each side with some matrix) didn't come to my mind.

Thank you again, you helped me much!