Chapter 7.2, Problem 34E

Finding a Power of a Matrix In Exercises 33-36, use the result of Exercise 31 to find the power of A shown.

$A=\left[\begin{array}{cc}1& 3\\ 2& 0\end{array}\right],A^7$

Proof Let A be a diagonalizable $n\times n$ matrix and let P be an invertible $n\times n$ matrix such that $B=P^{-1}AP$ is the diagonal form of A. Prove that $A^k=P{B}^k{P}^{-1}$, where k is a positive integer.