If `A = ({:(5, 5), (0, 0):})({:(0, 0), (5, 5):}) " and " A^(n) = ({:(5^(200), 5^(200)), (0, 0):})`, then find n.

Options

A. 100
B. 50
C. 25
D. None of these

Explanation

To solve the problem, we need to find the value of \( n \) such that \( A^n = \begin{pmatrix} 5^{200} & 5^{200} \\ 0 & 0 \end{pmatrix} \), given the matrix \( A = \begin{pmatrix} 5 & 5 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 \\ 5 & 5 \end{pmatrix} \).

### Step 1: Calculate the product of matrices A
We first need to multiply the two matrices to find \( A \).

\[ A = \begin{pmatrix} 5 & 5 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 \\ 5 & 5 \end{pmatrix} \]

Using the matrix multiplication formula, we calculate each element of the resulting matrix:

- First row, first column:
\[ (5 \cdot 0) + (5 \cdot 5) = 0 + 25 = 25 \]

- First row, second column:
\[ (5 \cdot 0) + (5 \cdot 5) = 0 + 25 = 25 \]

- Second row, first column:
\[ (0 \cdot 0) + (0 \cdot 5) = 0 + 0 = 0 \]

- Second row, second column:
\[ (0 \cdot 0) + (0 \cdot 5) = 0 + 0 = 0 \]

Thus, we have:
\[ A = \begin{pmatrix} 25 & 25 \\ 0 & 0 \end{pmatrix} \]

### Step 2: Calculate \( A^n \)
Next, we need to find \( A^n \). We can observe that \( A \) can be expressed as:
\[ A = 25 \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \]

Now, let's denote \( B = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \). Then:
\[ A^n = 25^n B^n \]

### Step 3: Find \( B^n \)
Now we need to find \( B^n \). We can calculate \( B^2 \) to see the pattern:
\[ B^2 = B \cdot B = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \]

It can be observed that \( B^n = B \) for any \( n \geq 1 \).

### Step 4: Compare \( A^n \) with given matrix
Now we have:
\[ A^n = 25^n B = 25^n \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \]

We need to compare this with:
\[ \begin{pmatrix} 5^{200} & 5^{200} \\ 0 & 0 \end{pmatrix} \]

This gives us:
\[ 25^n = 5^{200} \]

### Step 5: Solve for \( n \)
We know that \( 25 = 5^2 \), thus:
\[ (5^2)^n = 5^{200} \]
\[ 5^{2n} = 5^{200} \]

By equating the exponents, we have:
\[ 2n = 200 \implies n = \frac{200}{2} = 100 \]

### Final Answer
Therefore, the value of \( n \) is:
\[ \boxed{100} \]