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**Pinkk** Assuming that all matrices are $\displaystyle n\,\,x\,\,n$ and invertible, solve for $\displaystyle D$:

$\displaystyle ABC^{T}DBA^{T}C=AB^{T}$

I realized that I would someone have to reduce everything to the right of $\displaystyle D$ to $\displaystyle B^{T}I_{n}$ and everything to the left would be $\displaystyle AI_{n}$

So for the "right side", I started with $\displaystyle C^{-1}A^{-T}B^{-1} =(BA^{T}C)^{-1}$ since $\displaystyle C^{-1}A^{-T}B^{-1}BA^{T}C=I_{n}$ and therefore:

$\displaystyle B^{T}(BA^{T}C)^{-1}$ would result in $\displaystyle B^{T}I_{n}=B^{T}$

For the "left size", I started with $\displaystyle C^{-T}B^{-1}=(BC^{T})^{-1}$ since $\displaystyle ABC^{T}C^{-T}B^{-1}=AI_{n}=A$

Multiplying the "left" with the "right" to satisfy the whole equation, I ended up with:

$\displaystyle D=(BC^{T})^{-1}B^{T}(BA^{T}C)^{-1}$

Is this correct, or at the very least is my though process correct and if so, what possible errors are there? Thank you.