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Can someone please help me rearrange this matrix expression:

$$T=S(F^T)^{-1}=JF^{-1}\Sigma(F^T)^{-1}$$

I need $\Sigma$ in terms of all the other variables. I understand what to do when it is only $2$ variables but in which order do I multiply when it's more than $2$.

Thanks guys

Tricia
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1 Answers1

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If $J^{-1}$ exists (to show you the procedure).

\begin{align}JF^{-1}\Sigma \color{red}{(F^T)^{-1}}&=S\color{red}{(F^T)^{-1}}\\ JF^{-1}\Sigma \color{red}{(F^T)^{-1}}\color{blue}{F^T}&=S\color{red}{(F^T)^{-1}}\color{blue}{F^T}\\ \color{red}{J}F^{-1}\Sigma&=S\\ \color{blue}{J^{-1}}\color{red}{J}F^{-1}\Sigma&=\color{blue}{J^{-1}}S\\ \color{red}{F^{-1}}\Sigma&=J^{-1}S\\ \color{blue}F\color{red}{F^{-1}}\Sigma&=\color{blue}FJ^{-1}S\\ \Sigma&=\boxed{FJ^{-1}S}\end{align}