Let A be a 3 3 matrix with |A|=6.

Question

Let $A$ be a $3	imes 3$ matrix with $|A|=6$. Let $X=A^2\cdot\operatorname{adj}(2A)$. If $\big|\operatorname{adj}(3\,\operatorname{adj}(X))\big|=2^m\cdot 3^n$ for $m,n\in\mathbb{N}$, then $m+n$ is equal to \_\_\_\_.

Answer: 62

Accepted Answer

Since $A$ is $3	imes3$ with $|A|=6$: $|A^2|=|A|^2=36$.

$|2A|=2^3|A|=48$, and for a $3	imes3$ matrix $M$, $|\operatorname{adj}(M)|=|M|^{2}$, so $|\operatorname{adj}(2A)|=|2A|^2=48^2$.

Then $|X|=|A^2|\cdot|\operatorname{adj}(2A)|=36\cdot 48^2$. Writing $36=2^2\cdot3^2$ and $48^2=2^8\cdot 3^2$, we get $|X|=2^{10}\cdot 3^{4}$.

$|3\operatorname{adj}(X)|=3^3\cdot|\operatorname{adj}(X)|=27\cdot|X|^2=27\cdot(2^{10}\cdot3^4)^2=27\cdot2^{20}\cdot3^8=2^{20}\cdot3^{11}$.

Finally $|\operatorname{adj}(3\operatorname{adj}(X))| = |3\operatorname{adj}(X)|^2=(2^{20}\cdot 3^{11})^2=2^{40}\cdot3^{22}$.

So $m=40,\ n=22$, and $m+n=62$.

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