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Question
If \[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\]then An (where n ∈ N) equals
Options
\[\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\]
\[\begin{bmatrix}1 & n^2 a \\ 0 & 1\end{bmatrix}\]
\[\begin{bmatrix}1 & na \\ 0 & 0\end{bmatrix}\]
\[\begin{bmatrix}n & na \\ 0 & n\end{bmatrix}\]
Solution
\[\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\]
\[Here, \]
\[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 + 0 & a + a \\ 0 + 0 & 0 + 1\end{bmatrix} = \begin{bmatrix}1 & 2a \\ 0 & 1\end{bmatrix}\]
\[ A^3 = A^2 \times A = \begin{bmatrix}1 & 2a \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 + 0 & a + 2a \\ 0 + 0 & 0 + 1\end{bmatrix} = \begin{bmatrix}1 & 3a \\ 0 & 1\end{bmatrix} \]
This pattern is applicable for all natural numbers.
\[\therefore A^n = \begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\]
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