The solution to the matrix $A$ is as follows:

1. Characteristic Equation: $\lambda^3 - \lambda^2 - 5\lambda - 3 = 0$

2. Eigenvalues: $\lambda_1 = 3, \quad \lambda_2 = -1, \quad \lambda_3 = -1$

3. Eigenvectors:

   • Corresponding to $\lambda_1 = 3$: $\mathbf{v_1} = \begin{bmatrix} 0.408 \\ 0.408 \\ 0.816 \end{bmatrix}$
   • Corresponding to $\lambda_2 = -1$: $\mathbf{v_2} = \begin{bmatrix} 0.505 \\ 0.162 \\ 0.848 \end{bmatrix}$
   • Corresponding to $\lambda_3 = -1$: $\mathbf{v_3} = \begin{bmatrix} -0.334 \\ -0.911 \\ 0.243 \end{bmatrix}$

Would you like further details or clarifications?

Here are 5 related questions for further exploration:

1. How do you derive the characteristic equation of a matrix?
2. Can a matrix have repeated eigenvalues? What does that imply?
3. What are the properties of eigenvectors corresponding to distinct eigenvalues?
4. How do you normalize an eigenvector, and why is it important?
5. Can non-square matrices have eigenvalues and eigenvectors?

Tip: Eigenvalues and eigenvectors are often used in stability analysis, particularly in systems of differential equations.