The system of linear equationsx+λy−z=0λx−y−z=0x+y−λz=0has a non-trivial solution for :(1) exactly three values of λ.(2) infinitely many values of λ.(3) exactly one value of λ.(4) exactly two values of λ.
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take determinant of these equation
det { 1 lemda -1
lemda -1 -1
1 1 -lemda } =0
now ,
1 (lemda +1) -lemda (1-lemda^2)-1 (lemda+1) =0
=> lemda (1-lemda^2)=0
=> lemda = 0 , -1 , 1
hence option (1) is correct exactly three real value of lemda
det { 1 lemda -1
lemda -1 -1
1 1 -lemda } =0
now ,
1 (lemda +1) -lemda (1-lemda^2)-1 (lemda+1) =0
=> lemda (1-lemda^2)=0
=> lemda = 0 , -1 , 1
hence option (1) is correct exactly three real value of lemda
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