https://r2.erweima.ai/mathbot/image/1718341672733b5z1ak36.png

The image shows a determinant \( D_n \) of a matrix with a specific pattern:

\[
D_n = \begin{vmatrix}
2a & a^2 & 0 & \cdots & 0 \\
1 & 2a & a^2 & \cdots & 0 \\
0 & 1 & 2a & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & a^2 \\
0 & 0 & 0 & 1 & 2a \\
\end{vmatrix}
\]

To find the determinant of this matrix, we can use a recursive approach. Let's denote the determinant by \( D_n \). Notice the structure of the matrix: it is a tridiagonal matrix, where the main diagonal has \( 2a \), the subdiagonal has 1, and the superdiagonal has \( a^2 \).

The recursive formula for the determinant of such a tridiagonal matrix is given by:
\[
D_n = 2a D_{n-1} - a^2 D_{n-2}
\]
with initial conditions:
\[
D_1 = 2a
\]
\[
D_2 = \begin{vmatrix}
2a & a^2 \\
1 & 2a
\end{vmatrix} = (2a)(2a) - (a^2)(1) = 4a^2 - a^2 = 3a^2
\]

Now let's calculate a few initial values to see the pattern:

1. For \( n=1 \):
\[
D_1 = 2a
\]

2. For \( n=2 \):
\[
D_2 = 3a^2
\]

3. For \( n=3 \):
\[
D_3 = 2a D_2 - a^2 D_1 = 2a \cdot 3a^2 - a^2 \cdot 2a = 6a^3 - 2a^3 = 4a^3
\]

4. For \( n=4 \):
\[
D_4 = 2a D_3 - a^2 D_2 = 2a \cdot 4a^3 - a^2 \cdot 3a^2 = 8a^4 - 3a^4 = 5a^4
\]

5. For \( n=5 \):
\[
D_5 = 2a D_4 - a^2 D_3 = 2a \cdot 5a^4 - a^2 \cdot 4a^3 = 10a^5 - 4a^5 = 6a^5
\]

From these calculations, we observe that \( D_n \) follows the pattern:
\[
D_n = (n+1)a^n
\]

Thus, the general formula for the determinant \( D_n \) is:
\[
D_n = (n+1)a^n
\]

Would you like more details on the derivation, or do you have any other questions?

Here are some additional questions you might consider:
1. Can you show the step-by-step derivation of the recursive formula?
2. How does the structure of the matrix influence the determinant calculation?
3. Are there other methods to verify the determinant formula?
4. How can we apply this method to other tridiagonal matrices?
5. What are the practical applications of finding determinants of such matrices?

Tip: Recognizing patterns in matrix structures can simplify determinant calculations significantly.

Explore the recursive formula for finding the determinant of a tridiagonal matrix, with step-by-step derivation and general formula. This problem covers matrix theory, determinants, and recursive functions, suitable for undergraduate students.

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