Proof: Let A be invertible. Prove that $A^t$ is invertible and $(A^t)^{-1}=(A^{-1})^t$

Question

Let A be invertible. Prove that $A^t$ is invertible and $(A^t)^{-1}=(A^{-1})^t$

Let $A$ be a $m*n$ matrix, then $A^t$ is $n*m$ matrix.
I am wondering if this theorem works here.

Let V and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if $dim(V) = dim(W)$.

Am I suppose construct a mapping $T: V \rightarrow W$ where matrix $A \in V$ and $B \in W$ and prove that it is linear, injective and surjective?

Answer 1

$AA^{-1}=I$ implies that $(AA^{-1})^t=I^t=(A^{-1})^tA^t$ and $A^{-1}A=I$ implies that $(A^{-1}A)^t=I^t=I=A^t(A^{-1})^t$.