https://file.aiquickdraw.com/mathbot/file/17273438600822ldzcahv.jpg

The problem asks to determine the determinant of the matrix \( A(a) \), where:

\[
A(a) = \begin{pmatrix} 
1 & 0 & \ln a \\
0 & a & 0 \\
0 & 0 & 1 
\end{pmatrix}
\]

and \( a \in (0, +\infty) \).

We need to compute the determinant of \( A(e) \) where \( e \) is Euler's number (approximately 2.718). 

### Step 1: Determinant of \( A(a) \)
To calculate the determinant of a \( 3 \times 3 \) matrix, we use the formula:

\[
\det(A) = 1 \cdot \det \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} - 0 + \ln(a) \cdot \det \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}
\]

Simplifying the minor matrices:

- \( \det \begin{pmatrix} a & 0 \\ 0 & 1 \end{pmatrix} = a \times 1 - 0 = a \)
- \( \det \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = 0 \) (because there is a zero row)

Thus, the determinant of \( A(a) \) simplifies to:

\[
\det(A(a)) = 1 \cdot a + \ln(a) \cdot 0 = a
\]

### Step 2: Determinant of \( A(e) \)
Substitute \( a = e \) into the formula for \( \det(A(a)) \):

\[
\det(A(e)) = e
\]

### Conclusion:
The determinant of \( A(e) \) is \( e \), so the correct answer is:

\[
\boxed{e}
\]

Let me know if you have any further questions or need details on this explanation.

### Follow-up Questions:
1. How do you compute the determinant of a \( 2 \times 2 \) matrix?
2. What properties of logarithms might simplify matrix calculations?
3. Can the determinant of any matrix ever be negative?
4. How does the determinant help in determining whether a matrix is invertible?
5. What is the geometric interpretation of a matrix's determinant?

#### Tip: 
Always check if any rows or columns of the matrix contain zeros, as this can simplify determinant calculations significantly!

Se consideră matricea A(a) = (1 0 ln(a); 0 a 0; 0 0 1), unde a ∈ (0, +∞). Numarul det(A(e)) este egal cu?

Learn how to calculate the determinant of the matrix A(a) = (1 0 ln(a); 0 a 0; 0 0 1), where a ∈ (0, +∞), and evaluate it at a = e. This solution explores key linear algebra concepts, including the properties of determinants and logarithmic functions.

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