if A is a square matrix such that `A^(2)=A,` then det (A) is equal to

Options

A. 0 or 1
B. `-2 or 2`
C. `-3 or 3`
D. none of these

Answer

To solve the problem, we need to analyze the given condition that \( A^2 = A \). This means that \( A \) is an idempotent matrix. We will find the determinant of \( A \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ A^2 = A \]

2. **Rearrange the equation**:
\[ A^2 - A = 0 \]
This can be factored as:
\[ A(A - I) = 0 \]
where \( I \) is the identity matrix.

3. **Apply the determinant**:
Taking the determinant on both sides gives:
\[ \det(A(A - I)) = \det(0) \]
Since the determinant of the zero matrix is zero, we have:
\[ \det(A) \cdot \det(A - I) = 0 \]

4. **Analyze the product**:
The equation \( \det(A) \cdot \det(A - I) = 0 \) implies that at least one of the factors must be zero. Thus, we have two cases:
- \( \det(A) = 0 \)
- \( \det(A - I) = 0 \)

5. **Consider the second case**:
If \( \det(A - I) = 0 \), it means that \( A - I \) is singular, which implies that \( A \) has an eigenvalue of 1.

6. **Conclusion**:
Therefore, the determinant of \( A \) can either be:
- \( \det(A) = 0 \) (if \( A \) has no eigenvalue equal to 1)
- or \( \det(A) = 1 \) (if \( A \) is the identity matrix).

Thus, the possible values for \( \det(A) \) are:
\[ \det(A) = 0 \text{ or } 1 \]

### Final Answer:
The determinant of \( A \) is either \( 0 \) or \( 1 \).

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