If `A = [(1,0),(1/2,1)]` , then `A^100` is equal to

Options

A. `[(1,0),(25,0)]`
B. `[(1,0),(50,1)]`
C. `[(1,0),(1//2^(100),1)]`
D. none of these

Answer

To find \( A^{100} \) for the matrix \( A = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( A^2 \)
We start by multiplying matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 1 \cdot 1 + 0 \cdot \frac{1}{2} = 1 \)
- First row, second column: \( 1 \cdot 0 + 0 \cdot 1 = 0 \)
- Second row, first column: \( \frac{1}{2} \cdot 1 + 1 \cdot \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1 \)
- Second row, second column: \( \frac{1}{2} \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \)

Thus, we have:

\[ A^2 = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \]

### Step 2: Calculate \( A^3 \)
Now, we will calculate \( A^3 \) by multiplying \( A^2 \) by \( A \).

\[ A^3 = A^2 \cdot A = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 1 \cdot 1 + 0 \cdot \frac{1}{2} = 1 \)
- First row, second column: \( 1 \cdot 0 + 0 \cdot 1 = 0 \)
- Second row, first column: \( 1 \cdot 1 + 1 \cdot \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2} \)
- Second row, second column: \( 1 \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \)

Thus, we have:

\[ A^3 = \begin{pmatrix} 1 & 0 \\ \frac{3}{2} & 1 \end{pmatrix} \]

### Step 3: Identify the Pattern
From our calculations, we can see a pattern emerging:

- \( A^1 = \begin{pmatrix} 1 & 0 \\ \frac{1}{2} & 1 \end{pmatrix} \)
- \( A^2 = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \)
- \( A^3 = \begin{pmatrix} 1 & 0 \\ \frac{3}{2} & 1 \end{pmatrix} \)

It appears that the matrix \( A^n \) can be expressed as:

\[ A^n = \begin{pmatrix} 1 & 0 \\ \frac{n}{2} & 1 \end{pmatrix} \]

### Step 4: Calculate \( A^{100} \)
Using the pattern we identified, we can now calculate \( A^{100} \):

\[ A^{100} = \begin{pmatrix} 1 & 0 \\ \frac{100}{2} & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 50 & 1 \end{pmatrix} \]

### Final Answer
Thus, the result is:

\[ A^{100} = \begin{pmatrix} 1 & 0 \\ 50 & 1 \end{pmatrix} \]