\begin{aligned} & A^2=A A=\left(\begin{array}{cc} 3 & 4 \\ 4 & -3 \end{array}\right)\left(\begin{array}{cc} 3 & 4 \\ 4 & -3 \end{array}\right)=\left(\begin{array}{cc} 25 & 0 \\ 0 & 25 \end{array}\right)=\left(\begin{array}{cc} 5^2 & 0 \\ 0 & 5^2 \end{array}\right) \\ & A^3=A^2 A=\left(\begin{array}{cc} 5^2 & 0 \\ 0 & 5^2 \end{array}\right)\left(\begin{array}{cc} 3 & 4 \\ 4 & -3 \end{array}\right)=\left(\begin{array}{cc} 5^2 \times 3 & 5^2 \times 4 \\ 5^2 \times 4 & -5^2 \times 3 \end{array}\right) \\ & A^4=A^3 A=\left(\begin{array}{cc} 5^2 \times 3 \\ 5^2 \times 4 \\ 5^2 \times 4 \\ 4 & -3 \end{array}\right)=\left(\begin{array}{cc} 5^4 & 0 \\ 0 & 5^4 \\ 0 \end{array}\right) \\ & A^5=A^4 A=\left(\begin{array}{cc} 5^4 & 0 \\ 0 & 5^4 \end{array}\right)\left(\begin{array}{cc} 3 & 4 \\ 4 & -3 \end{array}\right)=\left(\begin{array}{cc} 5^4 \times 3 & 5^4 \times 4 \\ 5^4 \times 4 & -5^4 \times 3 \end{array}\right) \end{aligned}

\begin{aligned} & A^2=\left(\begin{array}{cc} 5^2 & 0 \\ 0 & 5^2 \end{array}\right), A^4=\left(\begin{array}{cc} 5^4 & 0 \\ 0 & 5^4 \end{array}\right), \cdots, A^{2 k}=\left(\begin{array}{cc} 5^{2 k} & 0 \\ 0 & 5^{2 k} \end{array}\right) \\ & A^3=\left(\begin{array}{cc} 5^2 \times 3 & 5^2 \times 4 \\ 5^2 \times 4 & -5^2 \times 3 \end{array}\right), A^5=\left(\begin{array}{cc} 5^4 \times 3 & 5^4 \times 4 \\ 5^4 \times 4 & -5^4 \times 3 \end{array}\right) \\ & \cdots, A^{2 k+1}=\left(\begin{array}{cc} 5^{2 k} \times 3 & 5^{2 k} \times 4 \\ 5^{2 k} \times 4 & -5^{2 k} \times 3 \end{array}\right) \end{aligned}
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