Question

The system of linear equations `x+lambday-z=0` `lambdax-y-z=0` `x+y-lambdaz=0` has a non-trivial solution for : (1) infinitely many values of `lambda` . (2) exactly one value of `lambda` . (3) exactly two values of `lambda` . (4) exactly three values of `lambda` .
A. Exactly one value of `lambda`
B. Exactly two values of `lambda`
C. Exactly three values of `lambda`
D. Infinitely many values of `lambda`

Answer 1

Correct Answer - 3
System has non-trivial solution
`:. |{:(1,,lambda,,-1),(lambda,,-1,,-1),(1,,1,,-lambda):}|=0`
`rAr 1(lambda +1) -lambda (-lambda^(2) +1) -1 (lambda +1) =0`
`rArr lambda +1 +lambda^(3)-lambda -lambda -1=0`
` rArr lambda^(3) -lambda =0`
`rArr lambda =0, +- 1`