If A is a non-singular matrix,

It is given that, $A$ is a non-singular matrix.

$\therefore|A| \neq 0$

$\Rightarrow\left|A^{\top}\right| \neq 0 \quad \quad\left(|A|=\left|A^{\top}\right|\right)$

Now.

$A A^{-1}=I_{n}=A^{-1} A$

$\Rightarrow\left(A A^{-1}\right)^{T}=\left(I_{n}\right)^{T}=\left(A^{-1} A\right)^{T}$

$\Rightarrow\left(A^{-1}\right)^{T} A^{T}=I_{n}=A^{T}\left(A^{-1}\right)^{T}$

$\Rightarrow\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$       (Using definition of inverse of a matrix)

If $A$ is a non-singular matrix, then $\left(A^{\top}\right)^{-1}=$    $\left(A^{-1}\right)^{T}$