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DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
By using properties of determinants, show that:
$\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}=(a+b+c)^3$​​
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Answer

$\text{L.H.S.}=\begin{vmatrix}a-b-c&2a&2a\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}$
$=\begin{vmatrix}a+b+c&a+b+c&a+b+c\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}\ \left[\text{R}_1\rightarrow\text{R}_1+\text{R}_2+\text{R}_3\right]$

$=(a+b+c)\begin{vmatrix}1&1&1\\2b&b-c-a&2b\\2c&2c&c-a-b\end{vmatrix}$

$=(a+b+c)\begin{vmatrix}1&0&0\\2b&-b-c-a&0\\2c&0&-c-a-b\end{vmatrix}\ \left[\text{C}_2\rightarrow\text{C}_2-\text{C}_1\ \text{and C}_3\rightarrow\text{C}_3-\text{C}_1\right]$


$=(a+b+c)1[(a+b+c)(a+b+c)-0]$

$=(a+b+c)(a+b+c)^2$

$=(a+b+c)^3=\text{R.H.S.}$

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