Let A and B are two `3xx3` non-zero real matrices and `AB=0`, then which of the following option is correct ?

Options

A. `AX=B` has unique solution
B. `AX=B` has infinite solutions
C. B is invertible
D. `adj(A)B` is invertible

Answer

To solve the problem, we need to analyze the condition given: \( AB = 0 \) where \( A \) and \( B \) are both \( 3 \times 3 \) non-zero real matrices.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We have two matrices \( A \) and \( B \) such that their product is the zero matrix. This means that the columns of \( B \) must lie in the null space of \( A \).

2. **Determinant Properties**:
We know that if \( AB = 0 \), then the determinant of the product can be expressed as:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) = 0 \]
Since \( A \) and \( B \) are non-zero matrices, at least one of the determinants must be zero.

3. **Analyzing Determinants**:
- If \( \text{det}(A) \neq 0 \), then \( A \) is invertible. However, if \( A \) is invertible, then \( AB = 0 \) implies \( B = 0 \), which contradicts the condition that \( B \) is a non-zero matrix.
- Therefore, we conclude that \( \text{det}(A) = 0 \).

4. **Conclusion about Matrix B**:
- Similarly, if \( \text{det}(B) \neq 0 \), then \( B \) would be invertible, leading to \( A = 0 \), which again contradicts the non-zero condition.
- Hence, we also conclude that \( \text{det}(B) = 0 \).

5. **Final Statement**:
Since both \( \text{det}(A) = 0 \) and \( \text{det}(B) = 0 \), both matrices \( A \) and \( B \) are singular. This means that there are infinitely many solutions for the equation \( AB = 0 \) since both matrices can be chosen from a set of singular matrices.

### Conclusion:
The correct option is that both \( A \) and \( B \) have determinants equal to zero, leading to infinitely many solutions.