If A and B are square matrices such that `A^(2020)=O and AB=A+B`, then `|B|` is equal to (where, O is a null matrix)

Options

A. 0
B. 1
C. `-1`
D. 4

Answer

To solve the problem, we need to analyze the given information about the square matrices \( A \) and \( B \).

### Step-by-Step Solution:

1. **Given Information**:
We know that:
\[ A^{2020} = O \]
where \( O \) is the null matrix. This implies that the matrix \( A \) is singular, meaning its determinant is zero:
\[ |A| = 0 \]

2. **Equation Relating A and B**:
We also have the equation:
\[ AB = A + B \]
Rearranging this equation gives:
\[ AB - A = B \]
or
\[ A(B - I) = 0 \]
where \( I \) is the identity matrix.

3. **Taking Determinants**:
We will take the determinant of both sides of the equation \( A(B - I) = 0 \):
\[ |A(B - I)| = |0| = 0 \]
Using the property of determinants, we know:
\[ |A(B - I)| = |A| \cdot |B - I| \]
Therefore, we have:
\[ |A| \cdot |B - I| = 0 \]

4. **Analyzing the Determinant**:
Since \( |A| = 0 \) (as established in step 1), the product \( |A| \cdot |B - I| = 0 \) does not provide information about \( |B - I| \). However, it indicates that either \( |A| = 0 \) or \( |B - I| = 0 \).

5. **Conclusion about Determinant of B**:
Since \( |A| = 0 \), we cannot conclude anything directly about \( |B| \) from this equation. However, we can analyze the implications:
- If \( A \) is singular, it does not necessarily mean \( B \) is non-singular.
- The equation \( A(B - I) = 0 \) suggests that \( B \) could also be singular.

6. **Final Result**:
Given that \( A \) is singular and the relationship \( AB = A + B \), it follows that \( |B| \) must also be zero to satisfy the equation under the condition that \( A \) is singular. Thus, we conclude:
\[ |B| = 0 \]

### Answer:
\[ |B| = 0 \]