Let M, N are two non-singular matrix of order 3 with real entries such that $adj(M) = 2N$ and $adj(N) = M$, then Prove that $MN=2I$
My approach is as follow
$adj\left( M \right) = 2N;adj\left( N \right) = M$
$adj\left( {adj\left( M \right)} \right) = adj\left( {2N} \right) = {\left| M \right|^{3 - 2}}M = 4adj\left( N \right) = 4M$
$\left| M \right| = 4$
$adj\left( {adj\left( N \right)} \right) = adj\left( M \right)$
${\left| N \right|^{3 - 2}}N = 2N \Rightarrow \left| N \right| = 2$
$MN = adj\left( N \right)N$
How do I proceed from here
Hint. $\operatorname{adj}(N)=|N|\cdot N^{-1}$.