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  4. The system of equations 3x - 2y+z=0, λ x -14y+ 15z=0, x + 2y-3z = 0 has a solution other than x=y = z = 0, then λ is equal to

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Q. The system of equations $3x - 2y+z=0$, $\lambda x -14y+ 15z=0$, $x + 2y-3z = 0$ has a solution other than $x=y = z = 0$, then $\lambda$ is equal to

Determinants

Solution:

The system of equations has infinitely many(non-trivial) solutions, i.e. $\Delta = 0$.
$\Rightarrow \quad\left|\begin{matrix}3&-2&1\\ \lambda&-14&15\\ 1&2&-3\end{matrix}\right|=0 \Rightarrow 3\left(42-30\right)-\lambda\left(6-2\right)+1\left(-30+14\right)=0$
$\Rightarrow \quad36-4\lambda - 16 = 0\, \Rightarrow \lambda= 5$