Question

If \(A=\begin{bmatrix}1&a\\0&1\end{bmatrix}\), then \(A^{k}\ (k\in\mathbf{N})\) is :

• A. \(\begin{bmatrix}1&ka\\0&1\end{bmatrix}\)
• B. \(\begin{bmatrix}1&k^{2}a\\0&1\end{bmatrix}\)
• C. \(\begin{bmatrix}k&ka\\0&k\end{bmatrix}\)
• D. \(\begin{bmatrix}1&ka\\0&0\end{bmatrix}\)

Hint:

Compute A squared and A cubed and spot the pattern; confirm by induction.

Answer 1

The Correct Option is A

\(A^2=\begin{bmatrix}1&a\\0&1\end{bmatrix}\begin{bmatrix}1&a\\0&1\end{bmatrix}=\begin{bmatrix}1&2a\\0&1\end{bmatrix}\).
\(A^3=A^2A=\begin{bmatrix}1&3a\\0&1\end{bmatrix}\).
Assume \(A^k=\begin{bmatrix}1&ka\\0&1\end{bmatrix}\); then \(A^{k+1}=A^kA=\begin{bmatrix}1&a+ka\\0&1\end{bmatrix}=\begin{bmatrix}1&(k+1)a\\0&1\end{bmatrix}\), so the pattern holds by induction.
Answer: (1) \(\begin{bmatrix}1&ka\\0&1\end{bmatrix}\)