1. a. Given \( A=\left[\begin{array}{ll}2 & 0 \\ 4 & 1\end{array}\right] \) compute the following,
i. \( A^{3} \)
ii. \( A^{-3} \)
iii. \( A^{2}-2 A+I \)
b. If a polynomial \( p(x) \) can be factored as a product of two lower degree polynomials \( p_{1}(x) \) and \( p_{2}(x) \) as
\[
p(x)=p_{1}(x) p_{2}(x)
\]
then it can be proved that \( p(A)=p_{1}(A) p_{2}(A) \) for an arbitrary square matrix \( A \). Verify this statement for polynomials
\[
p(x)=x^{2}-9, \quad p_{1}(x)=x+3, \quad p_{2}(x)=x-3
\]
