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# System

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Find all values of k for which the system

x + ky - z = 0

kx - y - z = 0

7x + 3y - 2kz = 0

has a non-trivial solution.  (In other words, find all values of k for which the system has a solution other than (x,y,z) = (0,0,0).)

May 14, 2022

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For a homogeneous system $$A \vec x = \vec 0$$ to have non-trivial solutions, we must have $$\det A = 0$$.

Then $$\det \begin{pmatrix}1&k&-1\\k&-1&-1\\7&3&-2k\end{pmatrix}= 0$$. Please simplify the determinant and solve the resulting cubic equation.

May 14, 2022
edited by MaxWong  May 14, 2022