First slide

System of linear equations

Question

The set of homoegeneous 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 non-trivial solution for

Moderate

A

Three values of $t$
B

Two values of $t$
C

One value of $t$
D

No value of $t$

Solution

Determinant of coefficients $= | \begin{matrix} t t + 1 t - 1 \\ t + 1 t t + 2 \\ t - 1 t + 2 t \end{matrix} | = | \begin{matrix} t 1 - 1 \\ t + 1 - 1 1 \\ t - 1 3 1 \end{matrix} |$ $= | \begin{matrix} t 1 - 1 \\ 2 t + 1 0 0 \\ 2 t - 1 4 0 \end{matrix} | = - 4 (2 t + 1)$ For non-trivial solution $t = - \frac{1}{2} .$

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