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(B) Given $A = \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bmatrix}$.Calculate $A^2 = A 	imes A = \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bma

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$\begin{bmatrix} 1 & 0 \ 25 & 1 \end{bmatrix}$

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(B) Given $A = \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bmatrix}$.Calculate $A^2 = A 	imes A = \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bmatrix} = \begin{bmatrix} 1(1) + 0(\frac{1}{2}) & 1(0) + 0(1) \ \frac{1}{2}(1) + 1(\frac{1}{2}) & \frac{1}{2}(0) + 1(1) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 2(\frac{1}{2}) & 1 \end{bmatrix}$.Calculate $A^3 = A^2 	imes A = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \ \frac{1}{2} & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 1 + \frac{1}{2} & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ \frac{3}{2} & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 3(\frac{1}{2}) & 1 \end{bmatrix}$.By induction,we can generalize that $A^n = \begin{bmatrix} 1 & 0 \ n(\frac{1}{2}) & 1 \end{bmatrix}$.Therefore,for $n = 50$,$A^{50} = \begin{bmatrix} 1 & 0 \ 50(\frac{1}{2}) & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 25 & 1 \end{bmatrix}$.

If $A = \begin{bmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{bmatrix}$,then $A^{50}$ is

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