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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
Using the property of determinants and without expanding, prove that:
$\begin{vmatrix}-a^{2}&ab&ac\\ba&-b^{2}&bc\\ca&cb&-c^{2}\end{vmatrix}=4a^2b^2c^2$
✓

Answer

$\text{L.H.S.}=\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}$
Taking common a,b,c from $R_1,R_2,R_3$ respectively,
$=abc\begin{vmatrix}-a&b&c\\a&-b&c\\a&b&-c\end{vmatrix}$
$=abc\begin{vmatrix}-a&b&c\\0&0&2c\\0&2b&0\end{vmatrix}\left[\text{operating}\ \text{R}_3\rightarrow\text{R}_3+\text{R}_1\ \text{and R}_2\rightarrow\text{ R}_2+\text{ R}_1\right]$
$=abc[-a(0-4bc)]$
$=abc(4abc)$
$=4a^2b^2c^2\ \text{R.H.S}$

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