The greatest value of c∈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

Given system of linear equations is x−cy−cz=0cx−y+cz=0 And cx+cy−z=0We know that a homogeneous system of linear equations have non-trivial solutions if Δ=0⇒1−c−cc−1ccc−1=0⇒11−c2+c−c−c2−cc2+c=0⇒ 1−c2−c2−c3−c3−c2=0⇒ −2c3−3c2+1=0⇒ 2c3−3c2−1=0⇒ (c+1)2c2+2c−c−1=0⇒ (c+1)(2c−1)(c−1)=0⇒ c=−1 or 12 Clearly _ the greatest value of c is 12.