If A and B are two matrices such that their product AB is
a null matrix, then

Options

A. det `A ne 0 rArr B` must be a null matrix
B. det `B ne 0 rArr A ` must be a null matrix
C. alteast one of the two matrices must be singular
D. if neither det A nor det B is zero, then the given statement
is not possible

Explanation

To solve the problem, we need to analyze the implications of the product of two matrices \( A \) and \( B \) being a null matrix (i.e., \( AB = 0 \)).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that the product of matrices \( A \) and \( B \) is a null matrix:
\[ AB = 0 \]

2. **Determinant of the Product**:
The determinant of the product of two matrices can be expressed as the product of their determinants:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
Since \( AB \) is a null matrix, we have:
\[ \text{det}(AB) = 0 \]

3. **Applying the Determinant Property**:
From the above, we conclude:
\[ \text{det}(A) \cdot \text{det}(B) = 0 \]
This implies that at least one of the determinants must be zero:
\[ \text{det}(A) = 0 \quad \text{or} \quad \text{det}(B) = 0 \]

4. **Interpreting the Result**:
If either \( \text{det}(A) = 0 \) or \( \text{det}(B) = 0 \), it indicates that at least one of the matrices \( A \) or \( B \) is singular (a singular matrix is one whose determinant is zero).

5. **Evaluating the Options**:
- Option 1: \( \text{det}(A} \neq 0 \) implies \( B \) must be a null matrix. (Incorrect)
- Option 2: \( \text{det}(B} \neq 0 \) implies \( A \) must be a null matrix. (Incorrect)
- Option 3: At least one of the two matrices must be singular. (Correct)
- Option 4: If neither \( \text{det}(A} \) nor \( \text{det}(B} \) is zero, then the given statement is not possible. (Correct)

### Conclusion:
The correct options are:
- Option 3: At least one of the two matrices must be singular.
- Option 4: If neither \( \text{det}(A} \) nor \( \text{det}(B} \) is zero, then the given statement is not possible.