If A is an invertible matrix of order 2 ; then det(A^-1) is equal to :

let $A=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]$

so $|A| = A_{11} A_{22}-A_{21} A_{12}$

finding cofactors of A

$C_{11}=A_{22} ~~C_{12}=-A_{21}$

$C_{21}=-A_{12} ~~C_{22}=A_{11}$

Adj A = $\left[\begin{array}{cc}A_{22} & -A_{21} \\ -A_{12} & A_{11}\end{array}\right]^T$

⇒  Adj  A = $\left[\begin{array}{cc}A_{22} & -A_{12} \\ -A_{21} & A_{11}\end{array}\right]$

$A^{-1}=\frac{1}{|A|} Adj~ A = \frac{1}{|A|}\left[\begin{array}{cc}
A_{22} & -A_{12} \\
-A_{21} & A_{11}
\end{array}\right]$

so $\left|A^{-1}\right|=\frac{1}{|A|^2}\left[A_{22} A_{11}-A_{12} A_{21}\right]$

$= \frac{|A|}{|A|^2}=\frac{1}{|A|}$