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 conditions step by step.

### Step-by-Step Solution:

1. **Given Conditions**:
We have two square matrices \( A \) and \( B \) such that:
\[ A^{2020} = O \quad \text{(where \( O \) is the null matrix)} \]
and
\[ AB = A + B. \]

2. **Determinant of \( A^{2020} \)**:
Since \( A^{2020} = O \), we can take the determinant of both sides:
\[ |A^{2020}| = |O|. \]
The determinant of the null matrix \( O \) is 0:
\[ |O| = 0. \]
Therefore,
\[ |A^{2020}| = 0. \]

3. **Using the Property of Determinants**:
We know that the determinant of a matrix raised to a power is equal to the determinant of the matrix raised to that power:
\[ |A^{2020}| = |A|^{2020}. \]
Thus, we have:
\[ |A|^{2020} = 0. \]

4. **Conclusion about \( |A| \)**:
The only way for \( |A|^{2020} = 0 \) is if \( |A| = 0 \). Therefore, we conclude:
\[ |A| = 0. \]

5. **Using the Second Condition \( AB = A + B \)**:
Now, we take the determinant of both sides of the equation \( AB = A + B \):
\[ |AB| = |A + B|. \]

6. **Applying the Determinant Product Rule**:
The determinant of a product of matrices is the product of their determinants:
\[ |AB| = |A| \cdot |B|. \]
Thus, we have:
\[ |A| \cdot |B| = |A + B|. \]

7. **Substituting \( |A| = 0 \)**:
Since we found that \( |A| = 0 \), we can substitute this into the equation:
\[ 0 \cdot |B| = |A + B|. \]
This simplifies to:
\[ 0 = |A + B|. \]

8. **Conclusion about \( |B| \)**:
The determinant \( |A + B| = 0 \) does not directly give us \( |B| \). However, since \( |A| = 0 \) and \( |AB| = |A + B| \), we can conclude that \( |B| \) must also be 0 in order for the equation to hold true.

Therefore, we conclude that:
\[ |B| = 0. \]

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