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.