4.176. $\frac{d x}{d t}=-2 x, \frac{d y}{d t}=-y$.
Solution. We form the characteristic equation

$$
\left|\begin{array}{cc}
(-2-\lambda) & 0 \\
0 & (-1-\lambda)
\end{array}\right|=(\lambda+1)(\lambda+2)=0
$$

The roots of the characteristic equation are real, distinct, and negative:

$$
\lambda_{1}=-1<0, \lambda_{2}=-2<0
$$

(case $1 \mathrm{a})-$ stable node.