For-what-values-of-a-does-the-following-system-have-a-non-trivial-solution-ax-1-3x-2-2x-3-0-x-1-4x-2-ax-3-0-5x-1-6x-2-7x-3-0-

Question Number 15232 by tawa tawa last updated on 08/Jun/17

For what values of a does the following system have a non trivial

solution

{ax}_{1} + {3 x}_{2} - {2 x}_{3} = 0

- x_{1} + {4 x}_{2} + {ax}_{3} = 0

{5 x}_{1} - {6 x}_{2} - {7 x}_{3} = 0

Answered by mrW1 last updated on 08/Jun/17

A = [\begin{matrix} a & 3 & - 2 \\ - 1 & 4 & a \\ 5 & - 6 & - 7 \end{matrix}]

∣ A ∣= a (- 28 + 6 a) - 3 (7 - 5 a) - 2 (6 - 20)

= {6 a}^{2} - 28 a - 21 + 15 a + 28

= {6 a}^{2} - 13 a + 7 = 0

a = \frac{13 \pm \sqrt{13^{2} - 4 \times 6 \times 7}}{2 \times 6} = \frac{13 \pm 1}{12} = 1, \frac{7}{6}

Commented by tawa tawa last updated on 08/Jun/17

God bless you sir .