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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Solve the following determinant equations:
$\begin{vmatrix}\text{x}+\text{a}&\text{b}&\text{c}\\\text{a}&\text{x}+\text{b}&\text{c}\\\text{a}&\text{b}&\text{x}+\text{c}\end{vmatrix}=0$
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Answer

Let $\begin{vmatrix}\text{x}+\text{a}&\text{b}&\text{c}\\\text{a}&\text{x}+\text{b}&\text{c}\\\text{a}&\text{b}&\text{x}+\text{c}\end{vmatrix}$
$=\begin{vmatrix}\text{x}+\text{a}+\text{b}+\text{c}&\text{b}&\text{c}\\\text{x}+\text{a}+\text{b}+\text{c}&\text{x}+\text{b}&\text{c}\\\text{x}+\text{a}+\text{b}+\text{c}&\text{b}&\text{x}+\text{c}\end{vmatrix} [$Applying $C_1 → C_1 + C_2 + C_3]$
$=(\text{x}+\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{b}&\text{c}\\1&\text{x}+\text{b}&\text{c}\\1&\text{b}&\text{x}+\text{c}\end{vmatrix}$
$=(\text{x}+\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{b}&\text{c}\\0&\text{x}&0\\1&\text{b}&\text{x}+\text{c}\end{vmatrix}$ $[$Applying $R_2 → R_2 - R_1]$
$=(\text{x}+\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{b}&\text{c}\\0&\text{x}&0\\1&0&\text{x}\end{vmatrix} [$Applying $R_3 → R_3 - R_1]$
$=(\text{x}+\text{a}+\text{b}+\text{c})(\text{x}^2-0)=0$ [Given]
$\Rightarrow\text{x}^2=0$ or $\text{x}+\text{a}+\text{b}+\text{c}=0$
$\Rightarrow\text{x}=0$ or $\text{x}=-(\text{a}+\text{b}+\text{c})$

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