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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS5 Marks
Question
Prove that:
$\begin{vmatrix} 1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}=\begin{vmatrix} 1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\\1&\text{c}&\text{c}^2\end{vmatrix}$
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Answer

$\begin{vmatrix} 1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$
Apply $R_1 \rightarrow R_1a, R_2 \rightarrow R_2b, R_3 \rightarrow R_3c$
$=\frac{1}{\text{abc}}\begin{vmatrix} \text{a}&\text{a}^2&\text{abc}\\\text{b}&\text{b}^2&\text{cab}\\\text{c}&\text{c}^2&\text{abc}\end{vmatrix}$
$=\frac{\text{abc}}{\text{abc}}\begin{vmatrix} \text{a}&\text{a}^2&1\\\text{b}&\text{b}^2&1\\\text{c}&\text{c}^2&1\end{vmatrix}$
$=-\begin{vmatrix} \text{a}&1&\text{a}^2\\\text{b}&1&\text{b}^2\\\text{c}&1&\text{c}^2\end{vmatrix}$
$=\begin{vmatrix} 1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\\1&\text{c}&\text{c}^2\end{vmatrix}$

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