How can we find the possible values of 'a' in a system of linear equations?

Question

What is/are the possible value(s) of $a$ such that the system of linear equations in $x$,$y$, and $z$ has a unique solution? $$\begin{align} x-2y+az&=0\\ y+3z&=-1\\ -2x+3y+z&=-2 \end{align}$$

Answer 1

Hint: write the system as $$\begin{pmatrix} 1 & -2 & a \\ 0 & 1 & 3 \\ -2 & 3 & 1 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ -1 \\ -2 \end{pmatrix}.$$ Now think about what the determinant can tell you about the matrix/the corresponding linear mapping.