If \(A=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}\), then the value of \(A^{20}\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}1 & 20\\0 & 1\end{bmatrix}\)
The first few powers show a pattern:
\( A^2 = \begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix} = \begin{bmatrix}1 & 2\\0 & 1\end{bmatrix} \)
\( A^3 = A^2A = \begin{bmatrix}1 & 2\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix} = \begin{bmatrix}1 & 3\\0 & 1\end{bmatrix} \)
In general, if \(A^n = \begin{bmatrix}1 & n\\0 & 1\end{bmatrix}\), then
\( A^{n+1} = A^nA = \begin{bmatrix}1 & n\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix} = \begin{bmatrix}1 & n+1\\0 & 1\end{bmatrix} \)
So each multiplication by A adds 1 to the top-right entry, and
\( A^{20} = \begin{bmatrix}1 & 20\\0 & 1\end{bmatrix} \)