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Question

If $$A=\begin{vmatrix} 0 & 0\\ 1 & 1\end{vmatrix}$$ then the value of $$A+A^2+A^3+....+A^n=?$$


A
A
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B
nA
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C
(n+1)A
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D
0
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Solution

The correct option is B nA
$${ A }^{ 2 }=A\times A=\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}$$
So , $${ A }^{ n }=A$$
Thus, $$\sum _{ r=1 }^{ n }{ { A }^{ r } } =nA$$

Mathematics

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