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Let A denote the matrix $\left(\begin{array}{ll}0 & i \\ i & 0\end{array}\right)$, where $i^{2}=-1$, and let I denote the identity matrix $\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$. Then $\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\ldots+\mathrm{A}^{2010} \mathrm{is}-$
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1730 Upvotes
Verified Answer
The correct answer is:
$\left(\begin{array}{ll}1 & 1 \\ \mathrm{i} & 1\end{array}\right)$
$$
\begin{array}{l}
\mathrm{A}=\left[\begin{array}{ll}
0 & \mathrm{i} \\
\mathrm{i} & 0
\end{array}\right] ; \mathrm{A}^{2}=\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right] ; \quad \mathrm{A}^{3}=\left[\begin{array}{cc}
0 & -\mathrm{i} \\
-\mathrm{i} & 0
\end{array}\right] ; \quad \mathrm{I}=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] \quad ; \quad \mathrm{A}^{4}=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]=\mathrm{I} \\
\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\ldots \ldots \mathrm{A}^{210} \\
\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}\right)+\mathrm{A}^{4}\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}\right)+\ldots+\mathrm{A}^{2008}\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}\right) \\
=\left(\begin{array}{ll}
1 & \mathrm{i} \\
\mathrm{i} & 1
\end{array}\right)
\end{array}
$$
\begin{array}{l}
\mathrm{A}=\left[\begin{array}{ll}
0 & \mathrm{i} \\
\mathrm{i} & 0
\end{array}\right] ; \mathrm{A}^{2}=\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right] ; \quad \mathrm{A}^{3}=\left[\begin{array}{cc}
0 & -\mathrm{i} \\
-\mathrm{i} & 0
\end{array}\right] ; \quad \mathrm{I}=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] \\
\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] \quad ; \quad \mathrm{A}^{4}=\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]=\mathrm{I} \\
\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}+\ldots \ldots \mathrm{A}^{210} \\
\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}\right)+\mathrm{A}^{4}\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}+\mathrm{A}^{3}\right)+\ldots+\mathrm{A}^{2008}\left(\mathrm{I}+\mathrm{A}+\mathrm{A}^{2}\right) \\
=\left(\begin{array}{ll}
1 & \mathrm{i} \\
\mathrm{i} & 1
\end{array}\right)
\end{array}
$$
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