If $A=\begin{bmatrix}1 & 0\\\frac{1}{25} & 1\end{bmatrix}$, then the value of $A^{125}$ is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\begin{bmatrix}1 & 0\\5 & 1\end{bmatrix}$
$A=\begin{bmatrix}1 & 0\\\frac{1}{25} & 1\end{bmatrix}$
$A^2=\begin{bmatrix}1 & 0\\\frac{2}{25} & 1\end{bmatrix}$
$A^3=\begin{bmatrix}1 & 0\\\frac{3}{25} & 1\end{bmatrix}$
so $A^n=\begin{bmatrix}1 & 0\\\frac{n}{25} & 1\end{bmatrix}⇒A^{125}\begin{bmatrix}1 & 0\\\frac{125}{25} & 1\end{bmatrix}$
$A^{125}\begin{bmatrix}1 & 0\\5& 1\end{bmatrix}$