[Math] Prove: If A is invertible, then adj(A) is invertible and $[\operatorname{adj}(A)]^{-1}=\frac1{\det(A)}A=\operatorname{adj}(A^{-1})$

Question

Prove: $\newcommand{\adj}{\operatorname{adj}}$If $A$ is invertible, then $\adj(A)$ is invertible and $[\adj(A)]^{-1}=\frac{1}{\det(A)}A=\adj(A^{-1})$

I can show the left side:

\begin{align*}
A^{-1}&=\frac{1}{\det(A)}\adj(A)\\
\implies AA^{-1}&=\frac{1}{\det(A)}A \cdot \adj(A)\\
\implies I&=\frac{1}{\det(A)}A\cdot \adj(A),
\end{align*}
and,
\begin{align*}
A^{-1}A&=\adj(A)\frac{1}{\det(A)}A\\
\implies I&=\adj(A)\frac{1}{\det(A)}A.
\end{align*}
So,
$$[\adj(A)]^{-1}=\frac{1}{\det(A)}A.$$

But I'm not sure how to show: $$\frac{1}{\det(A)}A=\adj(A^{-1}).$$

Answer 1

$$\newcommand{\adj}{\operatorname{adj}}A^{-1}=\frac{1}{\det(A)}\adj(A)$$ So,

\begin{align*} (A^{-1})^{-1}&=\frac{1}{\det(A^{-1})}\adj(A^{-1})\\ \iff A&=\frac{1}{\det(A^{-1})}\adj(A^{-1})\\ \iff \det(A^{-1})A &=\adj(A^{-1})\\ \iff \frac{1}{\det(A)}A&=\adj(A^{-1}). \end{align*}