The following equation is the general solution of a system of differential equations \( \mathbf{y}^{\prime}=A \mathbf{y} \). Select the type of eigenvalues \( A \) has. \( \left[\begin{array}{l}y_{1} \\ y_{2}\end{array}\right]=k_{1}\left[\begin{array}{c}2 \cos 4 t \\ -5 \cos 4 t-3 \sin 4 t\end{array}\right]+k_{2}\left[\begin{array}{c}2 \sin 4 t \\ -5 \sin 4 t+3 \sin 4 t\end{array}\right] \) \( \quad \) Select Real, distinct eigenvalues Repeated real eigenvalues Complex eigenvalues

The presence of trigonometric functions such as \( \cos(4t) \) and \( \sin(4t) \) in the solutions indicates that the eigenvalues of matrix \( A \) must be complex. Specifically, these functions arise from complex eigenvalues of the form \( \lambda = \alpha \pm i\beta \), where the imaginary part, \( \beta \), leads to oscillatory solutions. Hence, given that the coefficients involve \( 4t \), we deduce that the imaginary part of the eigenvalues is \( 4 \), confirming that \( A \) has complex eigenvalues. So, take a bow for recognizing the flair of oscillations in the solutions!