###### Activity4.3.4
1. Let's begin with the diagonal matrix

\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

Find the powers $$D^2\text{,}$$ $$D^3\text{,}$$ and $$D^4\text{.}$$ What is $$D^k$$ for a general value of $$k\text{?}$$

2. Suppose that $$A$$ is a matrix with eigenvector $$\vvec$$ and associated eigenvalue $$\lambda\text{;}$$ that is, $$A\vvec = \lambda\vvec\text{.}$$ By considering $$A^2\vvec\text{,}$$ explain why $$\vvec$$ is also an eigenvector of $$A$$ with eigenvalue $$\lambda^2\text{.}$$

3. Suppose that $$A= PDP^{-1}$$ where

\begin{equation*} D = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}

Remembering that the columns of $$P$$ are eigenvectors of $$A\text{,}$$ explain why $$A^2$$ is diagonalizable and find a diagonalization of it.

4. Give another explanation of the diagonalizability of $$A^2$$ by writing

\begin{equation*} A^2 = (PDP^{-1})(PDP^{-1}) = PD(P^{-1}P)DP^{-1}\text{.} \end{equation*}
5. In the same way, find a diagonalization of $$A^3\text{,}$$ $$A^4\text{,}$$ and $$A^k\text{.}$$

6. Suppose that $$A$$ is a diagonalizable $$2\times2$$ matrix with eigenvalues $$\lambda_1 = 0.5$$ and $$\lambda_2=0.1\text{.}$$ What happens to $$A^k$$ as $$k$$ becomes very large?

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