Let A=pmatrix2&1&01&2&-10&-1&2pmatrix.

Question

Let $A=\begin{pmatrix}2&1&0\1&2&-1\0&-1&2\end{pmatrix}$. If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(2A)))|=(16)^n$, then $n$ is equal to

Accepted Answer

$|A|=2(4-1)-1(2-0)=6-2=4$.

For a $3	imes3$ matrix, $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(2A)))|=|2A|^{(3-1)^3}=|2A|^8=(2^3|A|)^8=(2^3\cdot4)^8=(2^5)^8=2^{40}$.

$16^n=2^{4n}$, so $4n=40\Rightarrow n=10$.

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