1. Tardigrade
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  4. If the equations 2x + 3y + z = 0, 3x + y - 2z = 0 and ax + 2y - bz = 0 has non-trivial solution, then

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Q. If the equations $2x + 3y + z = 0$, $3x + y - 2z = 0$ and $ax + 2y - bz = 0$ has non-trivial solution, then

Determinants

Solution:

Since, the given equations has non-trival solution.
$\therefore \quad\left|\begin{matrix}2&3&1\\ 3&1&-2\\ a&2&-b\end{matrix}\right|=0$
$\Rightarrow \quad2\left(-b+4\right)-3 \left(-3b+2a\right)+1\left(6-a\right)=0$
$\Rightarrow \quad-2b+8+9b-6a+6-a=0$
$\Rightarrow \quad7b - 7a = - 14$
$\Rightarrow \quad a-b=2$