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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
If a, b and c are all non-zero and $\begin{vmatrix}1+\text{a}&1&1\\1&1+\text{b}&1\\1&1&1+\text{c} \end{vmatrix}=0,$ then prove that $\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}+1=0.$
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Answer

$\begin{vmatrix}1+\text{a}&1&1\\1&1+\text{b}&1\\1&1&1+\text{c} \end{vmatrix}=0$
$C_1 \rightarrow C_1 - C_2$
$\begin{vmatrix}\text{a}&1&1\\-\text{b}&1+\text{b}&1\\1&1&1+\text{c} \end{vmatrix}=0$
$C_2 \rightarrow C_2 - C_3​​​​​​​$​​​​​​​
$\begin{vmatrix}\text{a}&0&1\\-\text{b}&\text{b}&1\\0&-\text{c}&1+\text{c} \end{vmatrix}=0$
Expanding along $R_1,$ we get
$\text{a}(\text{b}+\text{bc}+\text{c})+1(\text{bc})=0$
$\Rightarrow\text{ab}+\text{abc}+\text{ac}+\text{bc}=0$
Dividing by abc, we get
$\frac{1}{\text{c}}+1+\frac{1}{\text{b}}+\frac{1}{\text{a}}=0$
$\therefore\frac{1}{\text{a}}+\frac{1}{\text{b}}+\frac{1}{\text{c}}+1=0$

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