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The question in your image asks about the determinant of an idempotent matrix \( A \), where \( A^2 = A \). It provides several options to choose from:

1. **Not enough information is given.**
2. **\( \det(A) \) can be either 1 or 0.**
3. **\( \det(A) \) is always positive.**
4. **\( \det(A) \) is always 1.**
5. **\( \det(A) \) is always 0.**

### Explanation:
An **idempotent matrix** \( A \) satisfies the condition \( A^2 = A \). This implies that the eigenvalues \( \lambda \) of the matrix must satisfy \( \lambda^2 = \lambda \), meaning the eigenvalues can only be 0 or 1.

Now, let's analyze the determinant:

- The **determinant** of a matrix is the product of its eigenvalues.
- Since the eigenvalues of an idempotent matrix can only be 0 or 1, the determinant of \( A \) can only be:
  - \( 0 \) if any of the eigenvalues is 0 (which would occur if \( A \) is not invertible).
  - \( 1 \) if all eigenvalues are 1 (in the special case where \( A \) is invertible).

### Conclusion:
The correct answer to this question is: **\( \det(A) \) can be either 1 or 0** (second option).

Would you like more details on eigenvalues and how they relate to determinants? Here are some related questions to deepen your understanding:

1. What are the properties of idempotent matrices?
2. How do eigenvalues determine the invertibility of a matrix?
3. Can an idempotent matrix be diagonalized?
4. How does the rank of an idempotent matrix relate to its eigenvalues?
5. What is the significance of the determinant being 0 or 1 for idempotent matrices?

**Tip:** Remember that eigenvalues offer a lot of insight into a matrix's properties, including its determinant and invertibility.

A square matrix A is said to be idempotent when A^2 = A. What can we say about the values of the determinant of an idempotent matrix A?

Understand the properties of an idempotent matrix A, where A^2 = A. Learn how the eigenvalues of A influence its determinant, which can be either 0 or 1. This explanation covers key concepts in linear algebra, including eigenvalues and determinants.

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