Question

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:
1. -1
2. \(\frac{1}{2}\)
3. 2
4. 0

Answer 1

Correct Answer - Option 2 : \(\frac{1}{2}\)

If the system of equations has non-trivial solutions, then the determinant of coefficient matrix is zero

\(\left| {\begin{array}{*{20}{c}} 1&{ - c}&{ - c}\\ c&{ - 1}&c\\ c&c&{ - 1} \end{array}} \right| = 0\)

No need to apply property just simplification is done.

⇒ 1(1 – c²) + c(– c – c²) – c(c² + c) = 0

⇒ (1 + c) (1 – c) – 2c²(1 + c) = 0

⇒ (1 + c) (1 – c – 2c²) = 0

⇒ (1 + c) (1 + c) (1 – 2c) = 0

⇒ (1 + c)² (1 – 2c) = 0

\({\rm{c}} = - 1{\rm{\;or\;}}\frac{1}{2}\)

Hence, the greatest value of \(c{\rm{\; = \;}}\frac{1}{2}\;\)for which the system of linear equations has non-trivial solution.