Question

​Solve the following determinant equation:

\[\begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix} = 0\]

 

Answer 1

\[\text{ Let }∆ = \begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix}\] 

\[ = \begin{vmatrix}x + 9 & 3 & 5 \\ x + 9 & x + 2 & 5 \\ x + 9 & 3 & x + 4\end{vmatrix} \left[\text{ Applying }C_1 = C_1 + C_2 + C_3 \right]\] 

\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 1 & x + 2 & 5 \\ 1 & 3 & x + 4\end{vmatrix} \] 

\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 1 & 3 & x + 4\end{vmatrix} \left[\text{ Applying }R_2 \text{ to } R_2 - R_1 \right]\] 

\[ = \left( x + 9 \right)\begin{vmatrix}1 & 3 & 5 \\ 0 & x - 1 & 0 \\ 0 & 0 & x - 1\end{vmatrix} \left[\text{ Applying }R_3 \text{ to } R_3 - R_1 \right]\] 

\[ ∆ = \left( x + 9 \right) \left( x - 1 \right)^2 = 0\] 

\[x = - 9, 1, 1\]