Question

# If $$A = \begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$$, then $$A^{n}$$ (where $$n\epsilon N$$) equals.

A
[1na01]
B
[1n2a01]
C
[1na00]
D
[nna0n]

Solution

## The correct option is A $$\begin{bmatrix}1 & na\\ 0 & 1\end{bmatrix}$$Given,$$A = \begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$$.Now,$$A^2 = \begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$$$$\begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$$or, $$A^2 = \begin{bmatrix}1 & 2a\\ 0 & 1\end{bmatrix}$$.Again,$$A^3 = \begin{bmatrix}1 & 2a\\ 0 & 1\end{bmatrix}$$$$\begin{bmatrix}1 & a\\ 0 & 1\end{bmatrix}$$or, $$A^3 = \begin{bmatrix}1 & 3a\\ 0 & 1\end{bmatrix}$$.Proceeding in this way using mathematical induction we've,$$A^n = \begin{bmatrix}1 & na\\ 0 & 1\end{bmatrix}$$.Maths

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