Let A be a 33 matrix such that |adj(adj(adjA))|=12^4.

Question

Let $A$ be a $3	imes3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))|=12^4$. Then $|A^{-1}\operatorname{adj}A|$ is equal to

Accepted Answer

For a $3	imes3$ matrix, $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))|=|A|^{(3-1)^3}=|A|^8$.

So $|A|^8=12^4=(2^2\cdot3)^4=2^8\cdot3^4$, giving $|A|=2\cdot3^{1/2}=2\sqrt3$ (taking the positive real root, since $2^8\cdot3^4=(2\sqrt3)^8$).

$|A^{-1}\operatorname{adj}A|=|A^{-1}|\cdot|\operatorname{adj}A|=\dfrac1{|A|}\cdot|A|^{3-1}=|A|=2\sqrt3$.

Solutions · 1