Prove that the following time-varying system is BIBO stable. $$ \begin{aligned} & \begin{bmatrix} \dot{x_1} \\ \dot{x_2} \end{bmatrix} = \begin{bmatrix} -t & 0 \\ 2t-1 & -2t \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} e^{-t} \\ e^{-2t} \end{bmatrix}u(t)\\ &\\ & \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} \cos(t) & \sin(t) \\ \sin(t) & -\cos(t) \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \end{aligned} $$
I tried to find the eigenvalues $\lambda_i(t)$ of $A(t)$ and they are indeed negative (i.e., $ \lambda_1(t) = -t $, $ \lambda_2(t) = -2t $ ) for $t\gt0$ but my textbook mentions that the eigenvalue method is only applicable to time-invariant systems.
If I consider $A$ static for any given time and show that its eigenvalues are negative for any $t\in T$, is that an acceptable proof?