If A= 1&a 0&1 , then A^n (where n N) equals: - Vidyadip

MCQ

If $\text{A}=\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix},$ then $A^n \ ($where $n \in N)$ equals$:$

✓
$\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$
B
$\begin{bmatrix}1&\text{n}^2\text{a}\\0&1\end{bmatrix}$
C
$\begin{bmatrix}1&\text{n}\text{a}\\0&0\end{bmatrix}$
D
$\begin{bmatrix}\text{n}&\text{n}\text{a}\\0&\text{n}\end{bmatrix}$

✓

Answer

Correct option: A.

$\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$

Given $\text{A}=\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$\text{A}^2=\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$=\begin{bmatrix}1&2\text{a}\\0&1\end{bmatrix}$
$\text{A}^3=\text{A}^2\text{A}$
$=\begin{bmatrix}1&2\text{a}\\0&1\end{bmatrix}\begin{bmatrix}1&\text{a}\\0&1\end{bmatrix}$
$=\begin{bmatrix}1&3\text{a}\\0&1\end{bmatrix}$
On genaralising we get
$\text{A}^\text{n}=\begin{bmatrix}1&\text{na}\\0&1\end{bmatrix}$

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