$w = aqe^{{\lambda_1}t} + cse^{{\lambda_2}t}$
$x = bqe^{{\lambda_1}t} + dse^{{\lambda_2}t}$
$y = are^{{\lambda_1}t} + cte^{{\lambda_2}t}$
$z = bre^{{\lambda_1}t} + dte^{{\lambda_2}t}$
EDIT:
To see why:
$(a x_0+by_0)\begin{bmatrix}q\\r\end{bmatrix} e^{\lambda_1t} + (cx_0+dy_0)\begin{bmatrix}s\\t\end{bmatrix}e^{\lambda_2t}$
$= [a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix}\begin{bmatrix}q\\r\end{bmatrix} e^{\lambda_1t} + [c\ d]\begin{bmatrix}x_0\\y_0\end{bmatrix}\begin{bmatrix}s\\t\end{bmatrix} e^{\lambda_1t}$
$= e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix} [a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix} + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix} [c\ d]\begin{bmatrix}x_0\\y_0\end{bmatrix}$ (this step works because $[a\ b]\begin{bmatrix}x_0\\y_0\end{bmatrix}$ for example is just a simple number)
$= (e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix}[a\ b] + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix}[c\ d])\begin{bmatrix}x_0\\y_0\end{bmatrix}$
So the matrix $\begin{bmatrix} w&x\\ y&z \end{bmatrix}$ you want is basically just $(e^{\lambda_1t}\begin{bmatrix}q\\r\end{bmatrix}[a\ b] + e^{\lambda_2t}\begin{bmatrix}s\\t\end{bmatrix}[c\ d])$.