Question

If  \[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\]then Aⁿ (where n ∈ N) equals 

 

Options

•  \[\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\] 

•  \[\begin{bmatrix}1 & n^2 a \\ 0 & 1\end{bmatrix}\] 

• \[\begin{bmatrix}1 & na \\ 0 & 0\end{bmatrix}\] 

•  \[\begin{bmatrix}n & na \\ 0 & n\end{bmatrix}\]

Answer 1

  \[\begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\] 

\[Here, \] 

\[A = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow A^2 = \begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 + 0 & a + a \\ 0 + 0 & 0 + 1\end{bmatrix} = \begin{bmatrix}1 & 2a \\ 0 & 1\end{bmatrix}\] 
\[ A^3 = A^2 \times A = \begin{bmatrix}1 & 2a \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & a \\ 0 & 1\end{bmatrix} = \begin{bmatrix}1 + 0 & a + 2a \\ 0 + 0 & 0 + 1\end{bmatrix} = \begin{bmatrix}1 & 3a \\ 0 & 1\end{bmatrix} \] 

This pattern is applicable for all natural numbers.

\[\therefore A^n = \begin{bmatrix}1 & na \\ 0 & 1\end{bmatrix}\]