Question Type 4: Rewriting 2x2 matrices using their diagonal matrix to calculate powers of a 2x2 matrix Exercises

Compute $A^9$ for the diagonal matrix $A=\begin{pmatrix}7 & 0 \\ 0 & 3\end{pmatrix}$.

Let $P=\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix},\quad D=\mathrm{diag}(3,1),\quad A=PDP^{-1}.$ Compute $A^8$.

Compute $A^6$ for $A=\begin{pmatrix}5 & 2 \\ 2 & 5\end{pmatrix}$ by diagonalization.

Compute $A^5$ for $A = \begin{pmatrix}4 & 3 \\ 2 & 1\end{pmatrix}$ using diagonalization.

Compute $A^4$ for $A=\begin{pmatrix}-1 & 2 \\ 2 & -1\end{pmatrix}$ by diagonalization.

Find $A^4$ for $A=\begin{pmatrix}3 & 4 \\ 2 & 1\end{pmatrix}$ using diagonalization.

Compute $A^7$ for $A=\begin{pmatrix}1 & 3 \\ 3 & 1\end{pmatrix}$ by diagonalization.

Find a general formula for $A^n$ where $A=\begin{pmatrix}2 & 1 \\ 1 & 2\end{pmatrix}$ by diagonalizing $A$.

Compute $A^5$ for $A=\begin{pmatrix}6 & -2 \\ 1 & 3\end{pmatrix}$ by diagonalizing $A$.

Find $A^{10}$ for $A=\begin{pmatrix}4 & 1 \\ 2 & 3\end{pmatrix}$ using diagonalization.

Find a general expression for $A^n$ where $A=\begin{pmatrix}4 & 3 \\ 1 & 2\end{pmatrix}$ by diagonalizing $A$.

For the symmetric matrix $A=\begin{pmatrix}a & b \\ b & a\end{pmatrix},$ derive a formula for $A^n$ in terms of $a,b,n$.