Solve the following determinant equations: 3x-8&3&3 3&3x-8&8 3&3&… - Vidyadip

Question

Solve the following determinant equations:
$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$

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Answer

$\begin{vmatrix}3\text{x}-8&3&3\\3&3\text{x}-8&8\\3&3&3\text{x}-8\end{vmatrix}=0$
Apply $C_1 \rightarrow C_1+ C_2 + C_3$
$\Rightarrow\begin{vmatrix}3\text{x}-2&3&3\\3\text{x}-2&3\text{x}-8&8\\3\text{x}-2&3&3\text{x}-8\end{vmatrix}=0$
$\Rightarrow(3\text{x}-2)\begin{vmatrix}1&3&3\\1&3\text{x}-8&8\\1&3&3\text{x}-8\end{vmatrix}=0$
$\Rightarrow(3\text{x}-2)\begin{vmatrix}1&3&3\\0&3\text{x}-11&0\\0&0&3\text{x}-11\end{vmatrix}=0$
$\Rightarrow(3\text{x}-2)(3\text{x}-11)^2=0$
$\Rightarrow(3\text{x}-2)=0$ or $(3\text{x}-11)^2=0$
$\Rightarrow\text{x}=\frac{2}{3}$ or $\text{x}=\pm\frac{11}{3}$

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