The set of all values of `lambda` for which the system of linear equations
`2x_(1)-2x_(2)+x_(3)=lambdax_(1)`
`2x_(1)-3x_(2)+2x_(3)=lambdax_(2)`
`-x_(1)+2x_(2)=lambdax_(3)`
has a non-trivial solution,

Options

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

Answer

A

Explanation

The given system of equations is
`x_(1) (2 - lamda) - 2x_(2) + x_(3) = 0`
`2x_(1) - (3 + lamda) x_(2) + 2x_(3) = 0`
`-x_(1) + 2x_(2) - lamda x_(3) = 0`
Clearly, it is a homogenous system of equations and will have non-trivial solution, if
`|(2 - lamda,-2,1),(2,-(3 + lamda),2),(-1,2,-lamda)| = 0`
`rArr lamda^(3) + lamda^(2) - 5 lamda + 3= 0`
`rArr lamda^(2) (lamda -1) + 2 lamda(lamda -1) - 3 (lamda -1) = 0`
`rArr (lamda -1) (lamda^(2) + 2 lamda -3) = 0`
`rArr (lamda -1) (lamda -1) (lamda + 3) = 0`
`rarr lamda = 1,1,3`
Hence, there are two values of `lamda`