A) \[\left[ \begin{matrix} 1 & n \\ 0 & 1 \\ \end{matrix} \right]\]
B) \[\left[ \begin{matrix} n & n \\ 0 & n \\ \end{matrix} \right]\]
C) \[\left[ \begin{matrix} n & 1 \\ 0 & n \\ \end{matrix} \right]\]
D) \[\left[ \begin{matrix} 1 & 1 \\ 0 & n \\ \end{matrix} \right]\]
Correct Answer: A
Solution :
\[{{A}^{2}}=\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]\,\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \\ \end{matrix} \right],\] and \[{{A}^{3}}={{A}^{2}}.A=\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \\ \end{matrix} \right]\,\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 3 \\ 0 & 1 \\ \end{matrix} \right]\] Þ \[{{A}^{n}}={{A}^{n-1}}.A=\left[ \begin{matrix} 1 & n-1 \\ 0 & 1 \\ \end{matrix} \right]\,\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & n \\ 0 & 1 \\ \end{matrix} \right]\].
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