The greatest value of `c in R` for which the system of linear equations x-cy-cz = 0, cx -y + cz =0, cx + cy-z =0 has a non-trivial solution, is

Options

A. `-1`
B. `(1)/(2)`
C. 2
D. 0

Explanation

Key Idea A homogeneous system of linear equations have non-trivial solutions iff `Delta = 0`
Given system of linear equations is
`x-cy-cz = 0`
`cx-y + cz =0`
`"and "cx + cy -z =0`
We know that a homogeneous system of linear equations have non-trivial solutions iff
`Delta = 0`
`rArr |{:(1, -c, -c), (c, -1, c), (c, c, -1):}| =0`
`rArr 1(1-c^(2)) +c(-c-c^(2)) -c(c^(2) +c) = 0`
`rArr 1-c^(2)-c^(2)-c^(3)-c^(3)-c^(2) =0`
`rArr -2c^(3)-3c^(2) +1 =0`
`rArr 2c^(3) + 3c^(2) -1 =0`
`rArr (c+1)[2c^(2) + c-1] =0`
`rArr (c+1)[2c^(2) +2c-c-1] = 0`
`rArr (c+1)(2c-1)(c+1)=0`
`rArr c = -1 "or" (1)/(2)`
Clearly, the greatest value of c is `(1)/(2)`.