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

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

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

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

Calculate $A^5$ for the matrix $A = \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}$ using the method of diagonalization.

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

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

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

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

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

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

Compute $\mathbf{A}^5$ by diagonalizing $\mathbf{A}$.

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