First slide

System of linear equations

Question

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

Moderate

A

three values of ‘ $t$ ’
B

two values of ‘ $t$ ’
C

one value of ‘ $t$ ’
D

Infinite number of values of ‘ $t$ ’

Solution

The condition for the system of equations to have a non-trivial solution is $| A | = 0$
$\begin{array}{l} \Rightarrow |\begin{matrix} t & t + 1 & t - 1 \\ t + 1 & t & t + 2 \\ t - 1 & t + 2 & t \end{matrix}| = 0, Apply R_{3} \to R_{3} - R_{2}; R_{2} \to R_{2} - R_{1} \\ \Rightarrow |\begin{matrix} t & t + 1 & t - 1 \\ 1 & - 1 & 3 \\ - 2 & 2 & - 2 \end{matrix}| = 0, Apply: C_{1} \to C_{1} + C_{2} \\ \Rightarrow |\begin{matrix} 2 t + 1 & t + 1 & t - 1 \\ 0 & - 1 & 3 \\ 0 & 2 & - 2 \end{matrix}| = 0 \\ \Rightarrow (2 t + 1) (2 - 6) = 0 \Rightarrow t = - 1 / 2 . \end{array}$

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