Using the property of determinants and without expanding, prove t… - Vidyadip

Question

Using the property of determinants and without expanding, prove that:
$\begin{vmatrix}1&bc&a(b+c)\\1&ca&b(c+a)\\1&ab&c(a+b)\end{vmatrix}=0$

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Answer

$\text{Given}:\ \begin{vmatrix}1&bc&a(b+c)\\1&ca&b(c+a)\\1&ab&c(a+b)\end{vmatrix}=\begin{vmatrix}1&bc&ab+ac\\1&ca&bc+ba\\1&ab&ca+cb\end{vmatrix}$
$\text{Operating}\ \text{C}_3\rightarrow\text{C}_3+\text{C}_2\ \begin{vmatrix}1&bc&ab+bc+ac\\1&ca&ab+bc+ca\\1&ab&ab+bc+ca\end{vmatrix}$
$=(ab+bc+ca)\begin{vmatrix}1&bc&1\\1&ca&1\\1&ab&1\end{vmatrix}$
$=(ab+bc+ca)(0)=0$ $\left[\because\text{two columns are identical Proved.}\right]$

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