For a matrix `A=[[1,2r-1] , [0,1]]` then `prod_(r=1)^(60) [[1,2r-1] , [0,1]]=`

Options

A. ` [[1, 100],[0,1]]`
B. ` [[1, 4950],[0,1]]`
C. ` [[1, 5050],[0,1]]`
D. ` [[1, 2500],[0,1]]`

Explanation

To solve the problem, we need to evaluate the product of the matrices \( A_r = \begin{bmatrix} 1 & 2r-1 \\ 0 & 1 \end{bmatrix} \) for \( r \) ranging from 1 to 60.

### Step-by-Step Solution:

1. **Define the Matrix**:
The matrix \( A_r \) is defined as:
\[ A_r = \begin{bmatrix} 1 & 2r - 1 \\ 0 & 1 \end{bmatrix} \]

2. **Calculate the Product**:
We need to compute the product:
\[ \prod_{r=1}^{60} A_r = A_1 \cdot A_2 \cdot A_3 \cdots A_{60} \]

3. **Find the First Few Matrices**:
- For \( r = 1 \):
\[ A_1 = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \]
- For \( r = 2 \):
\[ A_2 = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \]
- For \( r = 3 \):
\[ A_3 = \begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix} \]
- Continuing this pattern, we see that \( A_r \) has the form:
\[ A_r = \begin{bmatrix} 1 & 2r - 1 \\ 0 & 1 \end{bmatrix} \]

4. **Matrix Multiplication**:
The product of two matrices of the form \( A_i \) and \( A_j \) is:
\[ A_i \cdot A_j = \begin{bmatrix} 1 & 2i - 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2j - 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & (2i - 1) + (2j - 1) \\ 0 & 1 \end{bmatrix} \]
This shows that the upper right element of the resulting matrix is the sum of the upper right elements of the two matrices being multiplied.

5. **General Form of the Product**:
Continuing this multiplication for all \( r \) from 1 to 60, we find:
\[ \prod_{r=1}^{60} A_r = \begin{bmatrix} 1 & \sum_{r=1}^{60} (2r - 1) \\ 0 & 1 \end{bmatrix} \]

6. **Sum of Odd Numbers**:
The sum \( \sum_{r=1}^{60} (2r - 1) \) is the sum of the first 60 odd numbers, which is given by the formula:
\[ \text{Sum} = n^2 \quad \text{where } n \text{ is the number of terms} \]
Thus,
\[ \sum_{r=1}^{60} (2r - 1) = 60^2 = 3600 \]

7. **Final Result**:
Therefore, the product of the matrices is:
\[ \prod_{r=1}^{60} A_r = \begin{bmatrix} 1 & 3600 \\ 0 & 1 \end{bmatrix} \]

### Final Answer:
The final result of the product is:
\[ \begin{bmatrix} 1 & 3600 \\ 0 & 1 \end{bmatrix} \]