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 :
Solutions · 1
Official solution
If the system of equations has non-trivial
solutions, then
D = 0
\[\left| {1cc−c−1c−cc−1 } \right| = 0\]
\[ \Rightarrow \] (1 - c2) + c(-c - c2) - c(c2 + c) = 0
\[ \Rightarrow \] (1 + c)(1 - c - 2c2) = 0
\[ \Rightarrow \] (1 + c)2 (1 - 2c) = 0
\[ \Rightarrow \] c = -1 or \[{1 \over 2}\]
\[ \therefore \] Greatest value of c is \[{1 \over 2}\].