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 are given that \( A \) is a square matrix such that \( A^2 = A \). We need to find the value of \( \text{det}(A) \).

### Step-by-step Solution:

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

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

3. **Factor out \( A \):**
\[ A(A - I) = 0 \]
Here, \( I \) is the identity matrix of the same order as \( A \).

4. **Analyze the product:**
The equation \( A(A - I) = 0 \) implies that either \( A = 0 \) (the zero matrix) or \( A - I = 0 \) (which means \( A = I \)).

5. **Calculate the determinant in both cases:**
- If \( A = 0 \):
\[ \text{det}(A) = \text{det}(0) = 0 \]
- If \( A = I \):
\[ \text{det}(A) = \text{det}(I) = 1 \]

6. **Conclusion:**
Therefore, the determinant of \( A \) can be either \( 0 \) or \( 1 \):
\[ \text{det}(A) = 0 \text{ or } 1 \]