3. (1.5 points) Which of the following describes the equilibrium point of the following system of differential equations: \[ \frac{d Y}{d t}=\left(\begin{array}{cc}1 & 6 \\ 3 & -6\end{array}\right) Y \text {. } \] (a) center (b) sink (c) source (d) saddle (e) spiral sink (f) spiral source

To determine the type of equilibrium point for the given system of differential equations, we first analyze the coefficient matrix: \[\begin{pmatrix} 1 & 6 \\ 3 & -6 \end{pmatrix}.\] Calculating the eigenvalues gives us insights into the nature of the equilibrium point. The eigenvalues are found by solving the characteristic equation which is derived from the determinant of \(A - \lambda I\). In this case, we would get a calculation that leads to a positive and a negative eigenvalue, indicating the presence of a saddle point. Hence, the correct answer is (d) saddle. The behavior of systems like this has intrigued mathematicians and scientists for centuries! Systems of differential equations can predict everything from population dynamics to circuit behavior. These linear systems showcase how eigenvalues and eigenvectors provide crucial insight into stability and the trajectory of solutions, making them a cornerstone in both theoretical and applied mathematics.