Question

Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{det\left(adj\right(adj\left(adj\right(adjA\left)\right)\left)\right)}{(\alpha -\beta {)}^{16}(\beta -\gamma {)}^{16}(\gamma -\alpha {)}^{16}}=2^{32}\times 3^{16}$, then the number of such $3$-tuples $(\alpha ,\beta ,\gamma )$ is

• A. 42.0
• B. 42.00
• C. 42

Answer 1

Answer: (A), (B), (C)

$det\left(A\right)=\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right|$
$R_3\to R_3+R_1$
$\Rightarrow (\alpha +\beta +\gamma )\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ 1& 1& 1\end{array}\right|$

$\therefore det\left(A\right)=(\alpha +\beta +\gamma )(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )$
Also, $det\left(adj\right(adj\left(adj\right(adjA\left)\right)\left)\right)=\left(det\right(A){)}^{2^4}=(det\left(A\right){)}^{16}$

$\therefore \frac{(\alpha +\beta +\gamma {)}^{16}(\alpha -\beta {)}^{16}(\beta -\gamma {)}^{16}(\gamma -\alpha {)}^{16}}{(\alpha -\beta {)}^{16}(\beta -\gamma \left)\right(\gamma -\alpha {)}^{16}}=(4.3{)}^{16}$
$\Rightarrow \alpha +\beta +\gamma =12$
$\Rightarrow \alpha ,\beta ,\gamma$ distinct natural triplets
$={}^{11}{C}_2-1-{}^3{C}_2\left(4\right)$
$=55-1-12=42$