If A is a square matrix such that `A^(2)=A`, then |A| equals

Options

A. 0 or 1
B. 1 or 1
C. `-2` or 2
D. `-3` or 3

Answer

To solve the problem, we need to find the determinant of the square matrix \( A \) given that \( A^2 = A \).

### Step-by-Step Solution:

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

2. **Rearrange the equation:**
\[ A^2 - A = 0 \]

3. **Take the determinant of both sides:**
\[ |A^2 - A| = |0| \]

4. **Using the property of determinants:**
The determinant of a difference can be expressed as:
\[ |A^2 - A| = |A(A - I)| = 0 \]
where \( I \) is the identity matrix of the same order as \( A \).

5. **Apply the property of determinants:**
The determinant of a product of matrices is the product of their determinants:
\[ |A(A - I)| = |A| \cdot |A - I| = 0 \]

6. **Set up the equation:**
Since the product of the determinants equals zero, we have:
\[ |A| \cdot |A - I| = 0 \]

7. **Conclude the possible cases:**
This implies that either:
\[ |A| = 0 \quad \text{or} \quad |A - I| = 0 \]

8. **Analyze the cases:**
- If \( |A| = 0 \), then \( A \) is a singular matrix.
- If \( |A - I| = 0 \), then \( A \) has an eigenvalue of 1.

9. **Determine the possible values of \( |A| \):**
Since \( A^2 = A \) indicates that \( A \) is a projection matrix, the eigenvalues of \( A \) can only be 0 or 1. Therefore, the determinant \( |A| \), which is the product of the eigenvalues, must be either 0 or 1.

### Final Answer:
The possible values of \( |A| \) are:
\[ |A| = 0 \quad \text{or} \quad |A| = 1 \]