Matrices and Determinants 2 Question 26
29. Given that $x=-9$ is a root of $\begin{vmatrix}x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x\end{vmatrix}=0$, the other two roots are… and… .
(1983, 2M)
Show Answer
Answer:
Correct Answer: 29. (2 and 7)
Solution:
- Given, $\begin{vmatrix}x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x\end{vmatrix}=0$
Applying $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$
$ \Rightarrow\begin{vmatrix} x+9 & x+9 & x+9 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix} \Rightarrow 0 \Rightarrow(x+9)\begin{vmatrix} 1 & 1 & 1 \\ 2 & x & 2 \\ 7 & 6 & x \end{vmatrix}=0 $
Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $C_{3} \rightarrow C_{3}-C_{1}$
$ \begin{aligned} & \Rightarrow(x+9)\begin{vmatrix} 1 & 0 & 0 \\ 2 & x-2 & 0 \\ 7 & -1 & x-7 \end{vmatrix} \Rightarrow 0 \Rightarrow(x+9)(x-2)(x-7)=0 \\ & \Rightarrow \quad x = -9,2,7 \text{ are the roots.} \end{aligned} $
$\therefore$ Other two roots are 2 and 7 .
