Question

# If $\alpha ,\beta \text{ and }\gamma$ are the roots of the equation ${{x}^{3}}+px+q=0$ then the value of determinant $\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\\end{matrix} \right|$ is\begin{align} & A.p \\ & B.q \\ & C.{{p}^{2}}-2q \\ & D.0 \\ \end{align}

Hint: To solve this question, we will use three basic mathematical value which are given as below:
Sum of roots of a cubic (3 degree) equation is given by $\alpha +\beta +\gamma$ where equation is of the type $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$
We also know the formula, sum of roots $\Rightarrow \alpha +\beta +\gamma =\dfrac{-b}{a}=\dfrac{-\text{coefficient of }{{\text{x}}^{\text{2}}}}{\text{coefficient of }{{\text{x}}^{3}}}$
Using this, we will get the value of $\alpha +\beta +\gamma$ which can then be substituted in the determinant after applying row transformations.

We are given the equation as ${{x}^{3}}+px+q=0$.
Now, we know that sum of roots of a cubic (3 degree) equation can be written as $\alpha +\beta +\gamma$ where equation is of the type $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$. We also have a formula for finding the sum of roots. It is given as below,
$\Rightarrow \alpha +\beta +\gamma =\dfrac{-b}{a}=\dfrac{-\text{coefficient of }{{\text{x}}^{\text{2}}}}{\text{coefficient of }{{\text{x}}^{3}}}$
Here, in this given equation ${{x}^{3}}+px+q=0$ we do not have any term of ${{x}^{2}}$ so, the coefficient of ${{x}^{2}}=0$
Hence, we get the sum of roots of equation ${{x}^{3}}+px+q=0$ as
\begin{align} & \alpha +\beta +\gamma =0\text{ as coefficient of }{{\text{x}}^{\text{2}}}=0 \\ & \alpha +\beta +\gamma =0\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)} \\ \end{align}
Let us consider the determinant $\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|$
Determinant property says that, sum of elements of row or column changes the value of determinant. Hence, adding all rows element of above determinant we get:
$\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|=\left| \begin{matrix} \alpha +\beta +\gamma & \beta & \gamma \\ \alpha +\beta +\gamma & \gamma & \alpha \\ \alpha +\beta +\gamma & \alpha & \beta \\ \end{matrix} \right|$
Using equation (i) we get:
$\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|=\left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|$
Determinant property says that, if any one row or column is zero then the value of determinant is zero.
$\left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|=0$
Therefore, the value of $\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|=0$
Note: Another way to solve this question can be, opening the determinant $\left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|$ to get the required answer. Consider opening from first column, we get:
\begin{align} & \left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|=0\left| \begin{matrix} \gamma & \alpha \\ \alpha & \beta \\ \end{matrix} \right|-0\left| \begin{matrix} \beta & \gamma \\ \alpha & \beta \\ \end{matrix} \right|+0\left| \begin{matrix} \beta & \gamma \\ \gamma & \alpha \\ \end{matrix} \right| \\ & \Rightarrow \left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|=0-0-0 \\ & \Rightarrow \left| \begin{matrix} 0 & \beta & \gamma \\ 0 & \gamma & \alpha \\ 0 & \alpha & \beta \\ \end{matrix} \right|=0 \\ \end{align}
Hence, the value of determinants: $\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \\ \end{matrix} \right|=0$