If A = begin{bmatrix} 1 & 1 1 & 1 end{bmatri | vedclass

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(A) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.First,calculate $A^2 = A 	imes A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{b

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$2^{99} A$

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(A) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.First,calculate $A^2 = A 	imes A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1+1 & 1+1 \ 1+1 & 1+1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} = 2A$.Next,calculate $A^3 = A^2 	imes A = (2A) 	imes A = 2A^2 = 2(2A) = 2^2 A$.Similarly,$A^4 = A^3 	imes A = (2^2 A) 	imes A = 2^2 A^2 = 2^2(2A) = 2^3 A$.By induction,we can generalize that $A^n = 2^{n-1} A$ for any positive integer $n \geq 1$.Therefore,for $n = 100$,$A^{100} = 2^{100-1} A = 2^{99} A$.

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$,then $A^{100} = $ . . . . . . .

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