If $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$, then the value of $A^{20}$ is:

$\begin{bmatrix}1&20\\0&1\end{bmatrix}$

The correct answer is Option (3) → $\begin{bmatrix}1&20\\0&1\end{bmatrix}$

Given:

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

Matrix $A$ is an upper triangular matrix with eigenvalues 1 and 1.

Use the formula for powers of such matrix:

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

Therefore:

$A^{20} = \begin{bmatrix} 1 & 20 \\ 0 & 1 \end{bmatrix}$