Question

​Solve the following determinant equation:

\[\begin{vmatrix}15 - 2x & 11 - 3x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{vmatrix} = 0\]

Answer 1

Let Δ `=|(15-2x,11-3x,7-x),(11,17,14),(10,16,13)|=0`

`=>|(15-2x-14+2x,11-3x,7-x),(11-28,17,14),(10-26,16,13)|=0`   `["Applying"  C_1->C_1-2C_3]`

`=>|(1,11-3x,7-x),(-17,17,14),(-16,16,13)|=0`

`=>|(12-3x,4-2x,7-x),(0,3,14),(0,3,13)|=0`    `["Applying"  C_1->C_1+C_2  "and"  C_2->C_2-C_3]`

\[ \Rightarrow \left( 12 - 3x \right)\left( \left( 3 \times 13 \right) - \left( 3 \times 14 \right) \right) = 0\] 

\[ \Rightarrow \left( 12 - 3x \right)\left( - 3 \right) = 0\] 

\[ \Rightarrow 12 - 3x = 0\] 

\[ \Rightarrow 3x = 12\] 

\[ \Rightarrow x = 4\]