Question

If i=$\sqrt{-1}$ and $\sqrt[4]{41}=\alpha ,\beta ,\gamma ,\delta$ then $\left|\begin{array}{cccc}\alpha & \beta & \gamma & \delta \\ \beta & \gamma & \delta & \alpha \\ \gamma & \delta & \alpha & \beta \\ \delta & \alpha & \beta & \gamma \end{array}\right|$ is equal to

• A. $i$
• B. $-i$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

$\left|\begin{array}{cccc}\alpha & \beta & \gamma & \delta \\ \beta & \gamma & \delta & \alpha \\ \gamma & \delta & \alpha & \beta \\ \delta & \alpha & \beta & \gamma \end{array}\right|$
$C_1\to C_1+C_2+C_3+C_4$
$=\left|\begin{array}{cccc}\alpha +\beta +\gamma +\delta & \beta & \gamma & \delta \\ \alpha +\beta +\gamma +\delta & \gamma & \delta & \alpha \\ \alpha +\beta +\gamma +\delta & \delta & \alpha & \beta \\ \alpha +\beta +\gamma +\delta & \alpha & \beta & \gamma \end{array}\right|$
$=(\alpha +\beta +\gamma +\delta )\left|\begin{array}{cccc}1& \beta & \gamma & \delta \\ 1& \gamma & \delta & \alpha \\ 1& \delta & \alpha & \beta \\ 1& \alpha & \beta & \gamma \end{array}\right|$
Since, $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation
So, $\alpha +\beta +\gamma +\delta =0$
So, the value of determinant is 0.