Find the values of h for which the following set of vectors is linearly independent

Question

Find the values of h for which the following set of vectors is linearly independent

    1   h   1
    0   1   h
    0   h   1

Answer 1

Find the values of $h$ for which the determinant of the matrix $$\begin{pmatrix} 1 & h & 1 \\ 0 & 1 & h\\ 0&h&1\end{pmatrix}$$ is nonzero. This determinant is $1-h^{2}$, hence all values of $h$ are allowed except $h=1$ and $h=-1$.

Answer 2

solve the system $$\alpha+\beta h+\gamma=0$$ $$\beta+\gamma h=0$$ $$\beta h+\gamma=0$$