Question

​Solve the following determinant equation:

\[\begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix} = 0\]

 

Answer 1

\[\text{ Let }∆ = \begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix}\] 

\[ = \begin{vmatrix}3x - 2 & 3 & 3 \\ 3x - 2 & 3x - 8 & 3 \\ 3x - 2 & 3 & 3x - 8\end{vmatrix} \left[\text{ Applying }C_1 = C_1 + C_2 + C_3 \right]\] 

\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 1 & 3x - 8 & 3 \\ 1 & 3 & 3x - 8\end{vmatrix} \] 

\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 0 & 3x - 11 & 0 \\ 1 & 3 & 3x - 8\end{vmatrix} \left[\text{ Applying }R_2 \text{ to }R_2 - R_1 \right]\] 

\[ = \left( 3x - 2 \right)\begin{vmatrix}1 & 3 & 3 \\ 0 & 3x - 11 & 0 \\ 0 & 0 & 3x - 11\end{vmatrix} \left[\text{ Applying }R_3 \text{ to }R_3 - R_1 \right]\] 

\[ ∆ = \left( 3x - 2 \right) \left( 3x - 11 \right)^2 = 0\] 

\[x = \frac{2}{3}, \frac{11}{3}, \frac{11}{3}\]