Question

If \( A = \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} \), then \( A^{50} \) is:

• A. \( \begin{bmatrix} 1 & 0 \\ 0 & 50 \end{bmatrix} \)
• B. \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
• C. \( \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
• D. \( \begin{bmatrix} 1 & 25 \\ 0 & 1 \end{bmatrix} \)

Hint:

When performing matrix exponentiation, observe the pattern in the first few powers of the matrix to predict the result for higher exponents.

Answer 1

The Correct Option is D

The powers of matrix \( A \) are calculated as follows: \( A^2 = \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \).

Subsequently, \( A^4 \) is computed: \( A^4 = (A^2)^2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \).

Based on the observed pattern, it is inferred that: \( A^{50} = \begin{bmatrix} 1 & 0 \\ 25 & 1 \end{bmatrix} \).

As the computed result does not correspond to any of the provided options, the correct selection is "None of these."

Final Answer: \[ \boxed{\text{None of these}} \]