Find X for matrix $A(X-I)^{-1} = B^t$

Question

In an exam I had to find $X$ for $A(X-I)^{-1} = B^t$.

I did it this way: \begin{align} & A(X-I)^{-1}(X-I) = B^t(X-I) \\ \implies & A I = B^t(X-I) \\ \implies & A = B^t(X-I) \\ \implies & B^tA = B^tB^t(X-I) \\ \implies & B^{-1}B^tA = B^{-1}B(X-I) \\ \implies & B^{-1}B^tA = (X-I) \\ \implies & B^{-1}B^tA = X-I \\ \implies & B^{-1}B^tA+I = X \\ \implies & X = B^{-1}B^tA+I. \end{align} My teacher told me that this is wrong, but didn't give any explanation what did I do wrong.

From other students I found out that the right answer was $X = (A^{-1}B^t)^{-1}+I$

But if I will use this answer: \begin{align} & X = (A^{-1}B^t)^{-1}+I$ / (open $( )^{-1}$) \\ \implies & X = (B^t)^{-1}(A^{-1})^{-1}+I \\ \implies & X = (B^t)^{-1}A+I \\ \implies & X = (B^t)^{-1}A+I / (B^t)^{-1} = (B^{-1})^t \\ \implies & X = B^{-1}B^tA+I / (B^{-1})^t = B^{-1}B^t \\ \implies & X = B^{-1}B^tA+I. \end{align} And I get the same answer. Did I miss something, where is my mistake?

Answer 1

\begin{eqnarray} A(X-I)^{-1} &=& B^T \\ (X-I)^{-1} = A^{-1} B^T \\ X-I = (A^{-1}B^T)^{-1} \\ X = I+(A^{-1}B^T)^{-1} \\ \end{eqnarray}

Answer 2

Your mistake is $B^tB^t = B$ which you use in both derivations. $B^tB^t$ is not the same as $(B^t)^t$; and in general $B^tB^t \not= B$.

First here: $$ B^tA = B^tB^t(X-I) \\ \implies B^{-1}B^tA = B^{-1}B(X-I) $$

And a second time here: $$ \implies X = B^{-1}B^tA+I / (B^{-1})^t = B^{-1}B^t $$