Question Number 57619 by tanmay.chaudhury50@gmail.com last updated on 08/Apr/19
Answered by math1967 last updated on 09/Apr/19
$$\begin{vmatrix}{{q}+{r}}&{{r}+{p}}&{{p}+{q}}\\{{y}+{z}}&{{z}+{x}}&{{x}+{y}}\end{vmatrix} \\ $$$$\begin{vmatrix}{{b}+{c}−{c}−{a}−{a}−{b}}&{{c}+{a}}&{{a}+{b}}\\{{q}+{r}−{r}−{p}−{p}−{q}}&{{r}+{p}}&{{p}+{q}}\\{{y}+{z}−{z}−{x}−{x}−{y}}&{{z}+{x}}&{{x}+{y}}\end{vmatrix}{C}_{\mathrm{1}} −{C}_{\mathrm{2}} −{C}_{\mathrm{3}} \\ $$$$−\mathrm{2}\begin{vmatrix}{{a}}&{{c}+{a}}&{{a}+{b}}\\{{p}}&{{r}+{p}}&{{p}+{q}}\\{{x}}&{{z}+{x}}&{{x}+{y}}\end{vmatrix} \\ $$$$=−\mathrm{2}\begin{vmatrix}{{a}}&{{c}+{a}−{a}}&{{a}+{b}−{a}}\\{{p}}&{{r}+{p}−{p}}&{{p}+{q}−{p}}\\{{x}}&{{z}+{x}−{x}}&{{x}+{y}−{x}}\end{vmatrix}{C}_{\mathrm{2}} −{C}_{\mathrm{1}} ,{C}_{\mathrm{3}} −{C}_{\mathrm{1}} \\ $$$$=−\mathrm{2}\begin{vmatrix}{{a}}&{{c}}&{{b}}\\{{p}}&{{r}}&{{q}}\\{{x}}&{{z}}&{{y}\:}\end{vmatrix}=\mathrm{2}\begin{vmatrix}{{a}}&{{b}}&{{c}}\\{{p}}&{{q}}&{{r}}\\{{x}}&{{y}}&{{z}}\end{vmatrix}{C}_{\mathrm{2}} \Leftrightarrow{C}_{\mathrm{3}} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 09/Apr/19
$${thank}\:{you}\:{sir} \\ $$
Commented by math1967 last updated on 09/Apr/19
$${You}\:{are}\:{welcome}\:{sir} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 09/Apr/19
$${another}\:{way} \\ $$$$\mid{b}+{c}\:\:\:\:{c}+{a}\:\:\:{a}+{b}\:\mid \\ $$$$\mid{q}+{r}\:\:\:\:{r}+{p}\:\:\:\:{p}+{q}\mid \\ $$$$\mid{y}+{z}\:\:\:\:{z}+{x}\:\:\:\:{x}+{y}\mid \\ $$$$=\mid{b}\:\:{c}\:\:{a}\mid\:+\mid{b}\:\:{c}\:\:{b}\mid\:+\mid{b}\:\:\:{a}\:\:{a}\mid+\mid{b}\:\:{a}\:{b}\mid \\ $$$$\:\:\:\:\mid{q}\:\:{r}\:\:\:{p}\mid\:\:\:\:\:\mid{q}\:\:\:{r}\:\:\:{q}\mid\:\:\mid{q}\:\:\:\:{p}\:\:\:\:{p}\mid\:\:\:\mid{q}\:\:\:{p}\:\:{q}\mid\:\:+ \\ $$$$\:\:\:\:\mid\:{y}\:{z}\:{x}\mid\:\:\:\:\:\:\mid{y}\:\:\:{z}\:\:\:{y}\mid\:\:\mid{y}\:\:\:{x}\:\:\:\:{x}\mid\:\:\:\mid{y}\:\:{x}\:\:{y}\mid \\ $$$$ \\ $$$$\:\:\:\mid{c}\:\:\:{c}\:\:{a}\mid+\mid{c}\:\:{c}\:\:{b}\:\mid+\mid{c}\:\:{a}\:\:{a}\mid+\mid{c}\:\:{a}\:\:{b}\mid \\ $$$$\:\:\mid{r}\:\:\:{r}\:\:\:{p}\mid\:\:\:\mid{r}\:\:\:{r}\:\:{q}\mid\:\:\:\:\mid{r}\:\:\:{p}\:\:{p}\mid\:\:\:\mid{r}\:\:\:{p}\:\:\:{q}\mid \\ $$$$\:\:\:\mid{z}\:\:\:{z}\:\:\:{x}\mid\:\mid{z}\:\:\:\:{z}\:\:{y}\mid\:\:\:\mid{z}\:\:\:{x}\:\:\:{x}\mid\:\:\mid{z}\:\:\:{x}\:\:\:{y}\mid \\ $$$$ \\ $$$$\bigtriangleup=\bigtriangleup_{\mathrm{1}} +\bigtriangleup_{\mathrm{2}} +\bigtriangleup_{\mathrm{3}} +\bigtriangleup_{\mathrm{4}} +\bigtriangleup_{\mathrm{5}} +\bigtriangleup_{\mathrm{6}} +\bigtriangleup_{\mathrm{7}} +\bigtriangleup_{\mathrm{8}} \\ $$$${now}\:\bigtriangleup_{\mathrm{2}} =\bigtriangleup_{\mathrm{3}} =\bigtriangleup_{\mathrm{4}} =\bigtriangleup_{\mathrm{5}} =\bigtriangleup_{\mathrm{6}} =\bigtriangleup_{\mathrm{7}} =\mathrm{0} \\ $$$${reason}\:{two}\:{identical}\:{collumn} \\ $$$$ \\ $$$${so}\:\bigtriangleup=\bigtriangleup_{\mathrm{1}} +\bigtriangleup_{\mathrm{8}} \\ $$$${now}\:{inter}\:{change}\:{of}\:{collumn}\:{keep}\:{value}\:{of}\: \\ $$$${determinant}\:{unchanged} \\ $$$${So}\:\bigtriangleup=\mathrm{2}\mid{a}\:{b}\:{c}\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mid{p}\:\:{q}\:\:{r}\mid \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mid{x}\:\:\:{y}\:\:{z}\mid\:{proved} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$