8. (12 points) Let $A = \begin{bmatrix} 4 & -1 \\ 6 & -1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0.5 & 1.5 \end{bmatrix}$, and $C = \begin{bmatrix} 4 & -4.5 \\ 2 & -2 \end{bmatrix}$.

a. (6 points) Check the diagonalizability of each of the above matrices. If it is diagonalizable, please diagonalize it; otherwise, clearly explain why it is not.

b. (3 points) Find $A^8$ and $A^\infty$.

c. (3 points) Find $\det(B + 6I_3)$ using eigenvalues.