Let $A$ be a matrix of order $3 imes 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}$ then $\alpha+\beta+\gamma$ is equal to

Question

Let $A$ be a matrix of order $3 	imes 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}$ then $\alpha+\beta+\gamma$ is equal to

Aicharya · Accepted Answer

$\begin{aligned} & |2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))| \ & 2^3 \cdot|3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A})|^2 \ & 2^3 \cdot\left(3^3ight)^2 \cdot|\mathrm{~A}|^2 \cdot|\operatorname{adj}(2 \mathrm{~A})|^2 \ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2 \cdot\left(|2 \mathrm{~A}|^2ight)^2 \ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2\left[\left(2^3ight)^2 \cdot|\mathrm{~A}|^2ight]^2 \ & 2^3 \cdot 3^6 \cdot|\mathrm{~A}|^2 \cdot 2^{12} \cdot|\mathrm{~A}|^4 \ & 2^{15} \cdot 3^6 \cdot|\mathrm{~A}|^6 \ & 2^{15} \cdot 3^6 \cdot 5^6=2^\alpha \cdot 3^\beta \cdot 5^\gamma \ & \alpha=15, \quad \beta=6, \quad \gamma=6 \ & \alpha+\beta+\gamma=27\end{aligned}$ .

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A}…

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