The set of all values of `lambda` for which the system of linear equations: `x-2y-2z=lambdax`, `x+2y+z=lambday` and `-x-y=lambdaz` has a non-trivial solution: (a) is an empty sheet (b) is a singleton (c) contains more than two elements (d) contains exactly two elements

Options

A. is an empty set
B. contains exactly two elements
C. is a singleton
D. contains more than two elements

Answer

C

Explanation

Rearrange the equations
`x - lambday - 2y - 2z = 0, x + 2y - lambda y + z = 0`
`- x - y - lambda x = 0`
As `Delta_(1), Delta_(2), Delta_(3) = 0`, for non-trivial solution, `Delta = 0`
`Delta = |((1-lambda),-2,-2),(1,2-lambda,1),(-1,-1,-lambda)|`
`(1- lambda)[(-lambda)(2-lambda) + 1)] +2[-lambda + 1] -2[-1+1(2-lambda)] = 0`
`(1-lambda)[(lambda - 1)^(2) + 1] - 2[lambda-1] +2[lambda-1] = 0, (lambda-1)^(3) = 0 rArr lambda = 1`