Finding the value of θ using matrix algebra

Question

Determine the values of θ for which the system of equations,

\begin{align} x + y +z &= 1\\ x + 2y + 4z &= θ\\ x + 4y + 10z &= θ^2\\ \end{align}

possesses a solution, and for each such value find all solutions.

Answer 1

Note: $$\begin{vmatrix}1&1&1\\1&2&4\\1&4&10\end{vmatrix}=0.$$ Hence: $$\begin{vmatrix}1&1&1\\\theta&2&4\\\theta^2&4&10\end{vmatrix}=0 \Rightarrow \theta_{1,2}=1,2.$$