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### Problem Summary
1. **Matrix \( D \)**: It is a diagonal matrix with elements from 1 to 2000, i.e., \( D = \text{diag}(1, 2, 3, \dots, 2000) \).
2. **Matrix \( C \)**: It has mostly zero entries except for three specified values:
   - Element at row 1, column 2 is 5.
   - Element at row 499, column 499 is -8.
   - Element at row 500, column 501 is 3.
3. **Vector \( \vec{a} \)**: Defined as \( \vec{a} = (2^2, 2^3, \dots, 2^{2000})^T \).
4. **Matrix \( C + D \)**: We are to consider the multiplication \( (C + D) \vec{a} \).

Now, let’s go through each question in detail.

### Question (a)
**How many rows and columns does the matrix \( (C + D) \vec{a} \) have?**

Since \( D \) is a 2000x2000 diagonal matrix, \( C \) must also be a 2000x2000 matrix (to enable addition). Therefore, \( C + D \) is a 2000x2000 matrix. When we multiply \( (C + D) \) by \( \vec{a} \), the resulting matrix will have the dimensions of \( \vec{a} \), which is a 2000x1 vector.

- **Answer**: \( (C + D) \vec{a} \) has **2000 rows and 1 column**.

### Question (b)
**Determine the element in the 1st row and 1st column of \( (C + D) \vec{a} \).**

The entry in the 1st row and 1st column of \( (C + D) \vec{a} \) is given by:
\[
((C + D) \vec{a})_{1,1} = \sum_{j=1}^{2000} (C + D)_{1,j} \cdot a_j
\]
Since \( C \) is mostly zero except for specified entries, we only need to consider the elements affected by \( C \) and \( D \).

1. **Contribution from \( D \)**: Since \( D \) is diagonal, \( D_{1,1} = 1 \).
2. **Contribution from \( C \)**: Only \( C_{1,2} = 5 \) is non-zero in the first row.

So, we have:
\[
((C + D) \vec{a})_{1,1} = D_{1,1} \cdot a_1 + C_{1,2} \cdot a_2 = 1 \cdot 2^2 + 5 \cdot 2^3 = 4 + 5 \cdot 8 = 4 + 40 = 44
\]

- **Answer**: The element in the 1st row and 1st column of \( (C + D) \vec{a} \) is **44**.

### Question (c)
**Determine the element in the 500th row and 1st column of \( (C + D) \vec{a} \).**

The entry in the 500th row and 1st column of \( (C + D) \vec{a} \) is given by:
\[
((C + D) \vec{a})_{500,1} = \sum_{j=1}^{2000} (C + D)_{500,j} \cdot a_j
\]
Again, we only need to consider the non-zero elements in \( C \) and \( D \) for the 500th row.

1. **Contribution from \( D \)**: \( D_{500,500} = 500 \).
2. **Contribution from \( C \)**: The only non-zero element in row 500 of \( C \) is \( C_{500,501} = 3 \).

So, we have:
\[
((C + D) \vec{a})_{500,1} = D_{500,500} \cdot a_{500} + C_{500,501} \cdot a_{501} = 500 \cdot 2^{501} + 3 \cdot 2^{502}
\]
Simplify \( 2^{502} \) in terms of \( 2^{501} \):
\[
= 500 \cdot 2^{501} + 3 \cdot (2 \cdot 2^{501}) = 500 \cdot 2^{501} + 6 \cdot 2^{501} = (500 + 6) \cdot 2^{501} = 506 \cdot 2^{501}
\]

- **Answer**: The element in the 500th row and 1st column of \( (C + D) \vec{a} \) is \( 506 \cdot 2^{501} \).

---

Would you like additional details or further clarification on any of these steps?

### Related Questions:
1. How would the structure of \( D \) change if it were not a diagonal matrix?
2. How do we handle large exponent values in calculations involving matrices and vectors?
3. What would the result be if \( C \) had more non-zero entries?
4. Can the result in (c) be simplified further for computational efficiency?
5. How does the matrix \( C + D \) affect the vector \( \vec{a} \) compared to applying \( D \) alone?

### Tip:
When working with large powers in matrix computations, always look for ways to express terms in simpler forms using properties of exponents, which can make both manual and computational work easier.

Determine the dimensions of matrix (C + D)a, the first row, first column element, and the 500th row, first column element of (C + D)a, given matrices C, D, and vector a.

Explore a detailed solution to calculating the dimensions and specific elements of the matrix product (C + D)a, given matrices C (sparse), D (diagonal), and vector a with powers of two. This problem is ideal for understanding matrix operations, vector multiplication, and handling large powers of two.

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