It seems like your question might have some missing or unclear parts, particularly in "A[[23]] [[41]]." However, I'll assume you meant that matrix \( A \) is a \( 2 \times 2 \) matrix with elements 23 and 41. Let's clarify the matrix as follows:

\[
A = \begin{pmatrix} 23 & 41 \\ a & b \end{pmatrix}
\]

Here, \( a \) and \( b \) would typically be other elements in the matrix, but since they are not provided, I'll assume the matrix is:

\[
A = \begin{pmatrix} 23 & 41 \\ 23 & 41 \end{pmatrix}
\]

Given that, we need to find the matrix \( AA^t \), where \( A^t \) is the transpose of matrix \( A \).

### Steps to Solve

1. **Transpose of Matrix \( A \):**
   \[
   A^t = \begin{pmatrix} 23 & 23 \\ 41 & 41 \end{pmatrix}
   \]

2. **Matrix Multiplication \( AA^t \):**
   \[
   AA^t = \begin{pmatrix} 23 & 41 \\ 23 & 41 \end{pmatrix} \times \begin{pmatrix} 23 & 23 \\ 41 & 41 \end{pmatrix}
   \]

3. **Calculate \( AA^t \):**
   \[
   AA^t = \begin{pmatrix} (23 \times 23 + 41 \times 41) & (23 \times 23 + 41 \times 41) \\ (23 \times 23 + 41 \times 41) & (23 \times 23 + 41 \times 41) \end{pmatrix}
   \]

4. **Simplify the Expression:**
   \[
   AA^t = \begin{pmatrix} 529 + 1681 & 529 + 1681 \\ 529 + 1681 & 529 + 1681 \end{pmatrix} = \begin{pmatrix} 2210 & 2210 \\ 2210 & 2210 \end{pmatrix}
   \]

So, the final result for \( AA^t \) is:

\[
AA^t = \begin{pmatrix} 2210 & 2210 \\ 2210 & 2210 \end{pmatrix}
\]

### Questions:
1. What if the matrix \( A \) had different values for \( a \) and \( b \)?
2. How would \( AA^t \) differ if \( A \) was not symmetric?
3. What is the geometric interpretation of the matrix \( AA^t \)?
4. Can you generalize the process to an \( n \times n \) matrix?
5. What are the eigenvalues of the matrix \( AA^t \)?

### Tip:
When multiplying matrices, make sure that the dimensions match: the number of columns in the first matrix must equal the number of rows in the second matrix.

Explore how to compute AA^t for a 2x2 matrix with detailed steps. Learn about matrix multiplication, transposition, and their applications in linear algebra.

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