Menu Close

# If-and-are-roots-of-the-equation-x-3-px-q-0-p-0-q-0-then-find-the-value-of-the-determinant-determinant-

Question Number 363 by rajabhay last updated on 25/Jan/15
$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{and}\:\gamma\:\mathrm{are}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\$$$${x}^{\mathrm{3}} +{px}+{q}=\mathrm{0},\:{p}\neq\mathrm{0},\:{q}\neq\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the} \\$$$$\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{determinant}. \\$$$$\begin{vmatrix}{\alpha}&{\beta}&{\gamma}\\{\beta}&{\gamma}&{\alpha}\\{\gamma}&{\alpha}&{\beta}\end{vmatrix} \\$$
Commented by 123456 last updated on 24/Dec/14
$$\begin{vmatrix}{\alpha}&{\beta}&{\gamma}\\{\beta}&{\gamma}&{\alpha}\\{\gamma}&{\alpha}&{\beta}\end{vmatrix}=\mathrm{3}\alpha\beta\gamma−\left(\alpha^{\mathrm{3}} +\beta^{\mathrm{3}} +\gamma^{\mathrm{3}} \right) \\$$$${x}^{\mathrm{3}} +{px}+{q}=\mathrm{0} \\$$$$\alpha+\beta+\gamma=\mathrm{0} \\$$$$\alpha\beta+\alpha\gamma+\beta\gamma={p} \\$$$$\alpha\beta\gamma=−{q} \\$$
Answered by prakash jain last updated on 24/Dec/14
$$\begin{vmatrix}{\alpha}&{\beta}&{\gamma}\\{\beta}&{\gamma}&{\alpha}\\{\gamma}&{\alpha}&{\beta}\end{vmatrix}=\begin{vmatrix}{\alpha+\beta+\gamma}&{\beta}&{\gamma}\\{\beta+\gamma+\alpha}&{\gamma}&{\alpha}\\{\gamma+\alpha+\beta}&{\alpha}&{\beta}\end{vmatrix}\:\:\left({C}_{\mathrm{1}} ={C}_{\mathrm{1}} +{C}_{\mathrm{2}} +{C}_{\mathrm{3}} \right) \\$$$$=\begin{vmatrix}{\mathrm{0}}&{\beta}&{\gamma}\\{\mathrm{0}}&{\gamma}&{\alpha}\\{\mathrm{0}}&{\alpha}&{\beta}\end{vmatrix}=\mathrm{0} \\$$