Let A and B be two 33 matrices such that AB=I and |A|=18.

Question

Let $A$ and $B$ be two $3	imes3$ matrices such that $AB=I$ and $|A|=\dfrac18$. Then $|\operatorname{adj}(B\,\operatorname{adj}(2A))|$ is equal to

Accepted Answer

Since $AB=I$, $|A||B|=1\Rightarrow|B|=8$.

$|\operatorname{adj}(B\operatorname{adj}(2A))|=|B\operatorname{adj}(2A)|^2=|B|^2|\operatorname{adj}(2A)|^2=|B|^2\cdot|2A|^4=8^2\cdot(2^3\cdot	frac18)^4=64\cdot1^4=64.$

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