IB Mathematics Applications & Interpretation Question Type 4: Rewriting 2x2 Matrices Using Their Diagonal Matrix to Calculate Powers of a 2x2 Matrix Exercises

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

Find $A^4$ for $$A=\left(\begin{array}{cc}3& 4\\ 2& 1\end{array}\right)$$ using diagonalization.

Find a general formula for $A^n$ where $$A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$$ by diagonalizing $A$.

Calculate $A^5$ for the matrix $$A=\left(\begin{array}{cc}4& 3\\ 2& 1\end{array}\right)$$ using the method of diagonalization.

Compute $A^6$ for $$A=\left(\begin{array}{cc}5& 2\\ 2& 5\end{array}\right)$$ by diagonalization.

Compute $A^4$ for $$A=\left(\begin{array}{cc}-1& 2\\ 2& -1\end{array}\right)$$ by diagonalization.

Compute $A^9$ for the diagonal matrix $$A=\left(\begin{array}{cc}7& 0\\ 0& 3\end{array}\right)$$

Find $A^{10}$ for $$A=\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)$$ using diagonalization.

Compute $A^7$ for $$A=\left(\begin{array}{cc}1& 3\\ 3& 1\end{array}\right)$$ by diagonalization.

Let $$P=\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right),D=\text{diag}(3,1),A=PD{P}^{-1}.$$ Compute $A^8$.

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

Find a general expression for ${\mathbf{A}}^n$ where $$\mathbf{A}=\left(\begin{array}{cc}4& 3\\ 1& 2\end{array}\right)$$ by diagonalizing $\mathbf{A}$.