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Determinants — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Prove that:
$\begin{vmatrix}\text{a}+\text{b}+2\text{c}&\text{a}&\text{b}\\\text{c}&\text{b}+\text{c}+2\text{a}&\text{b}\\\text{c}&\text{a}&\text{c}+\text{a}+2\text{b} \end{vmatrix}=2(\text{a}+\text{b}+\text{c})^3$
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Answer

Let $\text{L.H.S}=\begin{vmatrix}\text{a}+\text{b}+2\text{c}&\text{a}&\text{b}\\\text{c}&\text{b}+\text{c}+2\text{a}&\text{b}\\\text{c}&\text{a}&\text{c}+\text{a}+2\text{b} \end{vmatrix}$
$=2(\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{a}&\text{b}\\1&\text{b}+\text{c}+2\text{a}&\text{b}\\1&\text{a}&\text{c}+\text{a}+2\text{b} \end{vmatrix}$ [Taking out 2(a + b + c) common from $C_1]$
$=2(\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{a}&\text{b}\\1&\text{b}+\text{c}+\text{a}&0\\0&-\text{b}-\text{c}-\text{a}&\text{c}+\text{a}+\text{b} \end{vmatrix}$ [Applying $R_2 \rightarrow R_2 - R_1$ and $R_2 \rightarrow R_2 - R_3]$
$=2(\text{a}+\text{b}+\text{c})(\text{a}+\text{b}+\text{c})(\text{a}+\text{b}+\text{c})\begin{vmatrix}1&\text{a}&\text{b}\\0&1&0\\0&-1&0\end{vmatrix}$ [Taking out (a + b + c) common from $R_2$​​​​​​​ and $R_3]$
$=2(\text{a}+\text{b}+\text{c})^3\{1(1-0)\}$ [Expanding along $C_1]$
$=2(\text{a}+\text{b}+\text{c})^3$
$=\text{R.H.S}$

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