Find $A^{-1}$ if $AB^TC = I_n$, where $A$, $B$, and $C$ are invertible $n \times n$ matrices

Question

Suppose $A$, $B$, and $C$ are invertible $n \times n$ matrices. Find $A^{-1}$ if $AB^TC = I_n$.

My answer is $A^{-1}B^{-T}C^{-1}$, but I am not sure if it is correct.

Answer 1

Obviously $$AB^TC=I_n$$ $$A^{-1}AB^TC=A^{-1}I_n$$ And at last $$A^{-1}=B^TC$$ You can do all of these operation because the three matrices are invertible, that is $A,B,C\in GL(n,\mathbb{K})$ where $\mathbb{K}$ is a field.

Moreover $AA^{-1}=I_n$ and $A^{-1}I_n=A^{-1}$.

Answer 2

$$AB^TC=I$$ Multiply both sides by $A^{-1}$ from left: $$A^{-1}AB^TC=A^{-1}I$$ Use the fact that $M^{-1}M=I$ and $MI=M$: $$IB^TC=A^{-1}$$ Use the fact that $IM=M$: $$B^TC=A^{-1}$$