If $$A = \begin{bmatrix}1 & x \\0 & 1\end{bmatrix}$$​ , then what is the value of $$A^n$$​ ? Where n is any natural number.

Correct option is C

Given:

​$$A = \begin{bmatrix}1 & x \\0 & 1\end{bmatrix}$$

​​Solution:

​$$A^1 = \begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix}$$

$$A^2 = A \cdot A = \begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \cdot 1 + x \cdot 0 & 1 \cdot x + x \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot x + 1 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 & 2x \\ 0 & 1 \end{bmatrix}$$

​​​$$A^3 = A^2 \cdot A = \begin{bmatrix} 1 & 2x \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \cdot 1 + 2x \cdot 0 & 1 \cdot x + 2x \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot x + 1 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 & 3x \\ 0 & 1 \end{bmatrix}$$

By continuing this pattern, we observe that:

$$A^n = \begin{bmatrix} 1 & nx \\ 0 & 1 \end{bmatrix}$$

Thus, the correct answer is (c).