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Consider the system of coupled linear differential equations:$$\begin{cases}\dfrac{dx}{dt} = 4x + 7y \\ \dfrac{dy}{dt} = -2x + 5y\end{cases}$$Let \(\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}\) and \(\dot{\mathbf{x}} = \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix}\).
Find the matrix \(\mathbf{A}\) such that \(\dot{\mathbf{x}} =\mathbf{A}\mathbf{x}\).
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