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).)
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.
