Question

The greatest value of c ε 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

• –1

• 2

• ${\displaystyle \frac{1}{2}}$

• 0

Answer 1

The greatest value of c ε 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 ${\displaystyle \underline{\frac{1}{2}}}$.

Explanation:

For non-trivial solution,

D = 0

${\displaystyle \left|\begin{array}{ccc}1& -c& -c\\ c& -1& c\\ c& c& -1\end{array}\right|}$ = 0

1(1 – c²) + c(–c – c²) –c(c² + c) = 0

1 – c² – c² – c³ – c³ – c² = 0

1 – 3c² – 2c³ = 0

2c³ + 3c² – 1 = 0

By remainder theorem

(c + 1)²(2c – 1) = 0

c = ${\displaystyle -1,\frac{1}{2}}$

Greatest value of c = ${\displaystyle \frac{1}{2}}$