First slide

System of linear equations

Question

The system of homogeneous equations , $(a - 1) x + (a + 2) y + a z = 0,$ $(a + 1) x + a y + (a + 2) z = 0$ , $a x + (a + 1) y + (a - 1) z = 0$
has a non-trivial solution if $a$ equals

Moderate

A

$\frac{1}{2}$
B

$- \frac{1}{2}$
C

2
D

-1

Solution

The system of equations will have a non-trivial solution if $∆ = 0$ where
$Δ = |\begin{matrix} a - 1 & a + 2 & a \\ a + 1 & a & a + 2 \\ a & a + 1 & a - 1 \end{matrix}|$
Using $C_{1} \to C_{1} - C_{3}$ and $C_{2} \to C_{2} - C_{3}$ , we obtain
$Δ = |\begin{matrix} - 1 & 2 & a \\ - 1 & - 2 & a + 2 \\ 1 & 2 & a - 1 \end{matrix}|$
Using $R_{3} \to R_{3} + R_{2}$ we get
$Δ = |\begin{matrix} - 1 & 2 & a \\ - 1 & - 2 & a + 2 \\ 0 & 0 & 2 a + 1 \end{matrix}| = (2 a + 1) (2 + 2) = 4 (2 a + 1)$
Now, $Δ = 0 \Rightarrow a = - 1 / 2$

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