Given a system of differential equations, how can one tell if $\textbf{x}_c = (0,0)^T$ is a unique critical point? I have: 
$$\frac{d\textbf{x}}{dt}=\begin{bmatrix}
    -1       & 2 \\
    -2       & -1  \\
\end{bmatrix}\textbf{x}(t)$$ with $\textbf{x}(0)=(1,-1)^T$.
I am asked whether the critical point $\textbf{x}_c=(0,0)^T$ is unique or not. 
I don't know how to go about answering this one.. Help please! :-)
Thanks!

The critical point(s) occur where
$\dfrac{d\mathbf x}{dt} = 0; \tag{1}$
since, as a vector field,
$\dfrac{d\textbf{x}}{dt}=\begin{bmatrix}
    -1       & 2 \\
    -2       & -1  \\
\end{bmatrix}\textbf{x}, \tag{2}$
the critical points occur wherever
$\begin{bmatrix}
    -1       & 2 \\
    -2       & -1  \\
\end{bmatrix}\textbf{x}_c = 0 \tag{3}$
has a solution.  We see that
$\det(\begin{bmatrix}
    -1       & 2 \\
    -2       & -1  \\
\end{bmatrix}) = 5 \ne 0; \tag{4}$
the coefficient matrix is thus non-singular; the only solution to (3) is
$\mathbf x_c = \begin{pmatrix} 0 \\ 0 \end{pmatrix}; \tag{5}$
thus this critical point is indeed unique.
