First slide

System of linear equations

Question

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

Moderate

A

$1$
B

$0$
C

$- 1$
D

None of these

Solution

For non-zero solution, $| \begin{matrix} a^{3} & {(a + 1)}^{3} & {(a + 2)}^{3} \\ a & a + 1 & a + 2 \\ 1 & 1 & 1 \end{matrix} | = 0$
$\Rightarrow - | \begin{matrix} 1 & 1 & 1 \\ a & a + 1 & a + 2 \\ a^{3} & {(a + 1)}^{3} & {(a + 2)}^{3} \end{matrix} | = 0$

$\Rightarrow - (a - a - 1) (a + 1 - a - 2) (a + 2 - a) \times (a + a + 1 + a + 2) = 0$

$[∵ | \begin{matrix} 1 & 1 & 1 \\ x & y & z \\ x^{3} & y^{3} & z^{3} \end{matrix} | = (x - y) (y - z) (z - x) (x + y + z)]$

$\Rightarrow - 2 (3 a + 3) = 0 \Rightarrow a = - 1$

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