Question
Let $$\alpha \,{\text{and }}\beta $$ be the roots of the equation $${x^2} + x + 1 = 0.$$ Then for $$y \ne 0\,\,{\text{in }}R,$$ \[\left| {\begin{array}{*{20}{c}}
{y + 1}&\alpha &\beta \\
\alpha &{y + \beta }&1\\
\beta &1&{y + \alpha }
\end{array}} \right|\] is equal to:
A.
$$y\left( {{y^2} - 1} \right)$$
B.
$$y\left( {{y^2} - 3} \right)$$
C.
$${{y^3}}$$
D.
$${{y^3} - 1}$$
Answer :
$${{y^3}}$$
Solution :
$${\text{Let}}\,\alpha \, = \omega \,\,{\text{and}}\,\beta = {\omega ^2} = 2$$ are roots of $${x^2} + x + 1 = 0$$
\[\& \,{\text{Let}}\,\Delta = \left| {\begin{array}{*{20}{c}}
{y + 1}&\omega &{{\omega ^2}}\\
\omega &{y + {\omega ^2}}&1\\
{{\omega ^2}}&1&{y + \omega }
\end{array}} \right| = \Delta \]
$${\text{Applying}}\,{C_1} \to {C_1} + {C_2} + {C_3},{\text{we}}\,{\text{get}}$$
\[\Delta {\rm{ = }}\left| {\begin{array}{*{20}{c}}
{y + 1 + \omega + {\omega ^2}}&\omega &{{\omega ^2}}\\
{y + 1 + \omega + {\omega ^2}}&{y + {\omega ^2}}&1\\
{1 + \omega + {\omega ^2} + y}&1&{y + \omega }
\end{array}} \right|\]
\[\Delta = \left| {\begin{array}{*{20}{c}}
y&\omega &{{\omega ^2}} \\
y&{y + {\omega ^2}}&1 \\
y&1&{y + \omega }
\end{array}} \right|\left( {\because \,\,1 + \omega + {\omega ^2} = 0} \right)\]
\[\Delta = y\left| {\begin{array}{*{20}{c}}
1&\omega &{{\omega ^2}}\\
1&{y + {\omega ^2}}&1\\
1&1&{y + \omega }
\end{array}} \right|\]
Applying $${R_2} \to {R_2} - {R_1}\,\,\& \,\,{R_3} \to {R_3} - {R_1},\,{\text{we get}}$$
\[\Delta = y\left| {\begin{array}{*{20}{c}}
{y + {\omega ^2} - \omega }&{1 - {\omega ^2}}\\
{1 - \omega }&{y + \omega - {\omega ^2}}
\end{array}} \right|\]
$$\eqalign{
& \Rightarrow \,\,\Delta = y \cr
& \left[ {y - \left( {\omega - {\omega ^2}} \right)\left( {y + \left( {\omega - {\omega ^2}} \right)} \right) - \left( {1 - \omega } \right)\left( {1 - {\omega ^2}} \right)} \right] \cr
& \Rightarrow \,\,\Delta = y\left[ {{y^2} - {{\left( {\omega - {\omega ^2}} \right)}^2} - 1 + {\omega ^2} + \omega - {\omega ^3}} \right] \cr
& \Rightarrow \,\,\Delta = y\left[ {{y^2} - {\omega ^2} - {\omega ^4} + 2{\omega ^3} - 1 + {\omega ^2} + {\omega ^4} - {\omega ^3}} \right]\,\,\left( {\because \,\,{\omega ^4} = \omega } \right) \cr
& \Rightarrow \,\,\Delta = y\left( {{y^2}} \right) = {y^3} \cr} $$