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

We start by calculating the powers of matrix \( A \): \[ A^2 = \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 1/2 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}. \]

Next, calculate \( A^4 \): \[ A^4 = (A^2)^2 = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}. \]

By observing the pattern, we deduce: \[ A^{50} = \begin{bmatrix} 1 & 0 \\ 25 & 1 \end{bmatrix}. \]

Since none of the given options match the result, the correct answer is "None of these."

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