<s>[INST] matrix-finding determinant of adj of inverse matirx if A is a $3$x$3$ matrix and let A=$2$,then what will be the value of det(adj(adj(adj($A^{-1}$)))?
1.$\dfrac{1}{512}$
2.$\dfrac{1}{1024}$
3.$\dfrac{1}{128}$
4.$\dfrac{1}{256}$
 [/INST] Assume you meant $det(A)=2$. Then $det(A^{-1})=\frac{1}{2}$. Work the problem outward:
\begin{align*}
adj(A^{-1})&=det(A^{-1})A=\frac{1}{2}A,\\
adj(\frac{1}{2}A)&=(1/2)^2 adj(A)=\frac{1}{4}det(A)A^{-1}=\frac{1}{2}A^{-1},\\
adj(\frac{1}{2}A^{-1})&=(1/2)^2adj(A^{-1})=\frac{1}{4}det(A^{-1})A=\frac{1}{8}A,\\
det(\frac{1}{8}A)&=(1/8)^3det(A)=\frac{1}{512}2=\frac{1}{256}.
\end{align*}
For justifications of the steps, you can wiki "adjugate matrix" and "determinant."
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