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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
Using the properties of determinants:
$\begin{vmatrix}0&\text{xy}^2&\text{xz}^2\\\text{x}^2\text{y}&0&\text{yz}^2\\\text{x}^2\text{z}&\text{zy}^2&0\end{vmatrix}$
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Answer

We have, $\begin{vmatrix}0&\text{xy}^2&\text{xz}^2\\\text{x}^2\text{y}&0&\text{yz}^2\\\text{x}^2\text{z}&\text{zy}^2&0\end{vmatrix}$
$=\text{x}^2\text{y}^2\text{z}^2\begin{vmatrix}0&\text{x}&\text{x}\\\text{y}&0&\text{y}\\\text{z}&\text{z}&0\end{vmatrix}$ $\big[\text{Taking x}^2,\text{y}^2,\text{z}^2\text{ common from C}_1,\text{C}_2\text{ and C}_3\big]$
$=\text{x}^2\text{y}^2\text{z}^2\begin{vmatrix}0&0&\text{x}\\\text{y}&-\text{y}&\text{y}\\\text{z}&\text{z}&0\end{vmatrix}$ $\big[\text{C}_2\rightarrow\text{C}_2-\text{C}_3\big]$
Expanding along $R_1$, we get
$=\text{x}^2\text{y}^2\text{z}^2[\text{x(yz}+\text{yz})]$
$=\text{x}^2\text{y}^2\text{z}^2.2\text{xyz}$
$=2\text{x}^3\text{y}^3\text{z}^3$

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