Assuming all matrices are nxn and invertible Solve for D ( inverses; algebraic properties of matrices) Is my answer right ?

Question

C^T B^-1 A² BAC^-1 DA^-2 B^T C^-2 = C^T

 

My answer is : D= BA^-2 B^-1 A^-1 CA² (B^T)^-1 C²

Answer 1

Let us do this step by step to isolate D and see what comes out.

 

C^T B^-1 A² BAC^-1 DA^-2 B^T C^-2 = C^T

 

Multiply the left of both sides by (C^TB^-1A² BAC^-1)^-1

 

DA^-2 B^T C^-2 = (C^TB^-1A² BAC^-1)^-1 C^T 

 

Multiply the right of both sides by A^-2 B^T C^-2

 

D =  [(C^TB^-1A² BAC^-1)^-1 C^T] A^-2 B^T C^-2

 

Simplify using the properties of inverses of matrix products, such as (MN)^-1 = N^-1M^-1

 

D = [CA^-1B^-1 A^-2 B(C^T)^-1 C^T] A^-2 B^T C^-2

 

D = CA^-1B^-1A^-2 B I A^-2 B^T C^-2

 

D = CA^-1B^-1A^-2 B A^-2 B^T C^-2

This is as far as I could simplify.

 

 

 

 

 

Answer 2

Nope. Not correct. I think you have the right idea that you have to "undo" all of the stuff on the lhs of the given equation, but you need to be careful also at the order in which you multiply from the left and right.