Find the value(s) of $a$ such that the given vectors span $\mathbb{R}^3$

Question

Considering the vectors $x=[1,1,-1]$, $y=[2,a+1,-1]$, $z=[1,0,-a]$

Find the conditions on a that ensures that

span$(x, y, z)$ = $\mathbb{R}^3$

Answer 1

You just have to make sure that the three vectors $x,y,z$ are indipendent.

There are several ways to do that, maybe the simpler one is just to impose that: $$\det{\begin{vmatrix}1 & 2 & 1 \\ 1 & a+1 & 0 \\ -1 & -1 & -a \end{vmatrix}} \neq 0$$