(Solved): ind the characteristic equation and the eigenvalues (and a basis for each of the corresponding eig ...

ind the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. \[ \left[\begin{array}{cc} -1 & -\frac{1}{2} \\ \frac{3}{2} & 1 \end{array}\right] \] (a) the characteristic equation (b) the eigenvalues (Enter your answers from smallest to largest.) \[ \left(\lambda_{1}, \lambda_{2}\right)=\left(\frac{-(1+\sqrt{6})}{2}, \frac{-1+\sqrt{6}}{2}\right) \] a basis for each of the corresponding eigenspaces \[ \mathbf{x}_{1}= \] \[ \mathbf{x}_{2}= \]