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DETERMINANTS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDETERMINANTS3 Marks
Question
Without expanding the determinant, prove that $\begin{vmatrix}a&a^2&bc\\b&b^2&ca\\c&c^2&ab\end{vmatrix}=\begin{vmatrix}1&a^2&a^3\\1&b^2&b^3\\1&b^2&b^3\end{vmatrix}$.
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Answer

$\text{L.H.S.}=\begin{vmatrix}a&a^2&bc\\b&b^2&ca\\c&c^2&ab\end{vmatrix}$
$=\frac{1}{abc}\begin{vmatrix}a^2&a^3&abc\\b^2&b^3&abc\\c^2&c^3&abc\end{vmatrix}\ \ \ \ \ \ \bigg[\text{R}_1\rightarrow a\text{R}_1,\text{R}_2\rightarrow b\text{R}_2,\text{and}\ \text{R}_3\rightarrow c\text{R}_3\bigg]$
$=\frac{1}{abc}.abc\begin{vmatrix}a^2&a^3&1\\b^2&b^3&1\\c^2&c^3&1\end{vmatrix}  [$Taking out factor abc from $C_3]$
$=\begin{vmatrix}a^2&a^3&1\\b^2&b^3&1\\c^2&c^3&1\end{vmatrix}$
$=\begin{vmatrix}1&a^2&a^3\\1&b^2&b^3\\1&c^2&c^3\end{vmatrix} [$Applying $C_1 \leftrightarrow C_3$ and $C_2 \leftrightarrow C_3 ]$
$=\text{R.H.S.}$
Hence, the given result is proved.

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