If the system of equations $$ax+y+z=0$$, $$x+by+z=0$$,$$x+y+cz=0$$,a,b,c≠$$1$$ has a non trivial solution $$($$ $$non-zero$$ $$solution)$$ , then $$11$$−$$a+11$$−$$b+11$$−$$c=$$

Question

If the system of equations $$ax+y+z=0$$, $$x+by+z=0$$,$$x+y+cz=0$$,a,b,c≠$$1$$  has a non trivial solution $$($$ $$non-zero$$ $$solution)$$ , then $$11$$−$$a+11$$−$$b+11$$−$$c=$$

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Given system of equations has a nontrivial solution   ⇒ coefficient matrix is singular⇒$$a111b111c=$$⇒abc−$$1$$−$$1c$$−$$1+11$$−$$b=0$$⇒abc−a−$$c+1+1$$−$$b=0$$⇒$$abc=a+b+c$$−$$211$$−$$a+11$$−$$b+11$$−$$c=(1$$−$$b)(1$$−$$c)+(1$$−$$a)(1$$−$$c)+(1$$−$$a)(1$$−$$b)(1$$−$$a)(1$$−$$b)(1$$−$$c)$$⇒$$3$$−$$2(a+b+c)+bc+ac+ab1$$−$$(a+b+c)+ab+bc+ca$$−$$abc=3$$−$$2(a+b+c)+bc+ca+ab1$$−$$(a+b+c)+ab+bc+ca$$−$$(a+b+c$$−$$2)=1$$

If the system of equations $a x + y + z = 0, x + b y + z = 0, x + y + c z = 0, (a, b, c \neq 1)$ has a non trivial solution ( non-zero solution) , then $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =$

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If the system of equations ax+y+z=0, x+by+z=0,x+y+cz=0,a,b,c≠1 has a non trivial solution ( non-zero solution) , then 11−a+11−b+11−c=

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If the system of equations $a x + y + z = 0, x + b y + z = 0, x + y + c z = 0, (a, b, c \neq 1)$ has a non trivial solution ( non-zero solution) , then $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =$