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The system of linear equations x + lam...

The system of linear equations
`x + lambda y-z =0, lambdax-y -z =0, x + y -lambda z =0`
has a non-trivial solution for

A

infinitely many values of `lambda`

B

exactly one value of `lambda`

C

exactly two values of `lambda`

D

exactly three values of `lambda`

Text Solution

Verified by Experts

The correct Answer is:
D
Given, system of linear equation is `x + lambday-z =0, lambdax-y-z =0, x+y-lambda z = 0`
Note that, given system will have a non-trivial solution only if determinant of coefficient matrix is zero,
`i.e. |{:(1, lambda, -1), (lambda, -1, -1), (1, 1, -lambda):}| =0`
`rArr 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(lambda^(2)-1) = 0`
`rArr lambda = "or" lambda =+-1`
Hence, given system of linear equation has anon-trivial solution for exactly three values of `lambda`.
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