First slide

System of linear equations

Question

The value of a for which the system of equations; $a^{3} {x + (a +1)}^{3} {y + (a + 2)}^{3} z = 0 , ax + (a + 1) y + (a + 2) z = 0 & x + y + z = 0$ has a non-zero solution is :

Moderate

A

1
B

0
C

-1
D

None of these

Solution

Use $C_{2} \to C_{1} & C_{3} \to C_{3} - C_{2}$ then open by $R_{3}$
$|\begin{matrix} a^{3} & {(a + 1)}^{3} & {(a + 2)}^{3} \\ a & a + 1 & a + 2 \\ 1 & 1 & 1 \end{matrix}| C_{2} \to C_{2} - C_{1,} C_{3} \to C_{3} - C_{2} |\begin{matrix} a^{3} & 3 a^{2} + 3 a + 1 & 3 a^{2} + 9 a + 7 \\ a & 1 & 1 \\ 1 & 0 & 0 \end{matrix}| |\begin{matrix} a^{3} & 3 a^{2} + 3 a + 1 & 6 a + 7 \\ a & 1 & 0 \\ 1 & 0 & 0 \end{matrix}| = - (6 a + 7) = 0$

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