The set of all values of λ for which the system of linear equations $$2x1$$−$$2x2+x3=$$λ$$x1$$,$$2x1$$−$$3x2+2x3=$$λ$$x2$$ and −$$x1+2x2=$$λ$$x3$$ has a $$non-trivial$$ solution

Question

The set of all values of  λ for which the system of linear equations $$2x1$$−$$2x2+x3=$$λ$$x1$$,$$2x1$$−$$3x2+2x3=$$λ$$x2$$  and  −$$x1+2x2=$$λ$$x3$$ has a $$non-trivial$$ solution

Infinity Learn · Accepted Answer

Given system of linear equations ⇒(2$$−λ$$)x1$$−$$2x2+x3=0----1$$ $$2x1$$−$$3x2+2x3=$$λ$$x2----2$$ ⇒$$2x1$$−$$(3+$$λ$$)x2+2x3=0----3$$ −$$x1+2x2=$$λ$$x3$$ ⇒−$$x1+2x2$$−λ$$x3=0Since$$ , the system has $$non-trivial$$ solution∴$$2$$−λ−$$212$$−$$(3+$$λ$$)2$$−$$12$$λ$$=0$$⇒$$(2$$−λ$$)3$$λ$$+$$λ$$2$$−$$4+2($$−$$2$$λ$$+2)+1(4$$−$$3)−λ$$=0$$⇒(2$$−λ$$)λ$$2+3$$λ−$$4+4(1$$−λ$$)+(1$$−λ$$)=0$$⇒ (2$$−λ$$)($$λ$$+4)($$λ−$$1)+5(1$$−λ$$)=0$$⇒$$($$λ−$$1)[(2$$−λ$$)($$λ$$+4)−$$5$$]$$=0$$⇒$$($$λ−$$1)$$λ$$2+2$$λ−$$3=0$$⇒$$($$λ−$$1)$$[$$($$λ−$$1)($$λ$$+8)$$]$$=0$$⇒$$($$λ−$$1)2($$λ$$+3)=0$$⇒ λ$$=1$$,$$1$$,−$$3Hence$$ , λ contains two elements.

The set of all values of $λ$ for which the system of linear equations $2 x_{1} - 2 x_{2} + x_{3} = λ x_{1}, 2 x_{1} - 3 x_{2} + 2 x_{3} = λ x_{2}$ and $- x_{1} + 2 x_{2} = λ x_{3}$ has a non-trivial solution

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The set of all values of λ for which the system of linear equations 2x1−2x2+x3=λx1,2x1−3x2+2x3=λx2 and −x1+2x2=λx3 has a non-trivial solution

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The set of all values of $λ$ for which the system of linear equations $2 x_{1} - 2 x_{2} + x_{3} = λ x_{1}, 2 x_{1} - 3 x_{2} + 2 x_{3} = λ x_{2}$ and $- x_{1} + 2 x_{2} = λ x_{3}$ has a non-trivial solution