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

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(B) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.Calculate $A^2$:$A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1

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$2^{99}A$

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(B) Given $A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}$.Calculate $A^2$:$A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1+1 & 1+1 \ 1+1 & 1+1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} = 2 \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = 2^1 A$.Calculate $A^3$:$A^3 = A^2 \cdot A = (2A) \cdot A = 2 A^2 = 2(2A) = 4A = 2^2 A$.By induction,we can generalize for $A^n$:$A^n = 2^{n-1} A$.Therefore,for $n = 100$:$A^{100} = 2^{100-1} A = 2^{99} A$.Thus,the correct option is $B$.

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

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