Let A be a 33 matrix of order 3 with |A|=5.

Question

Let $A$ be a $3	imes3$ matrix of order 3 with $|A|=5$. If $|2\operatorname{adj}(3A\,\operatorname{adj}(2A))|=2^\alpha\cdot3^\beta\cdot5^\gamma$, $\alpha,\beta,\gamma\in\mathbb{N}$, then $\alpha+\beta+\gamma$ is equal to:

Accepted Answer

$|2\operatorname{adj}(3A\,\operatorname{adj}(2A))|=2^3\big|3A\,\operatorname{adj}(2A)\big|^2=2^3(3^3)^2|A|^2\big|\operatorname{adj}(2A)\big|^2.$

For a $3\times3$ matrix, $|\operatorname{adj}(2A)|=|2A|^2=(2^3|A|)^2$, so $\big|\operatorname{adj}(2A)\big|^2=(2^3)^4|A|^4$.

$$=2^3\cdot3^6\cdot5^2\cdot(2^3)^4\cdot5^4=2^3\cdot3^6\cdot5^2\cdot2^{12}\cdot5^4=2^{15}\cdot3^6\cdot5^6.$$

So $\alpha=15,\ \beta=6,\ \gamma=6$, and $\alpha+\beta+\gamma=27$.

Solutions · 1