A, B, C are three non-zero matrices such that ABC = O, which of the following statements is true ?

Options

A. `|B|=0,|A| ne 0,|C|ne0`
B. `|B|ne 0,|C|=0,|A|ne0`
C. `|C|ne 0,|B|ne0,|A|=0`
D. none of these

Answer

To solve the problem, we need to analyze the given condition that the product of three non-zero matrices A, B, and C equals the zero matrix (O). We will also explore the implications of this condition on the determinants of these matrices.

### Step-by-Step Solution:

1. **Understanding the Condition**:
We are given that \( ABC = O \), where \( O \) is the zero matrix. This means that the product of matrices A, B, and C results in a matrix where all entries are zero.

2. **Properties of Determinants**:
Recall that for any square matrices \( A \), \( B \), and \( C \), the determinant of the product of matrices is equal to the product of their determinants:
\[ \text{det}(ABC) = \text{det}(A) \cdot \text{det}(B) \cdot \text{det}(C) \]

3. **Applying the Condition**:
Since \( ABC = O \), we have:
\[ \text{det}(ABC) = \text{det}(O) = 0 \]
Therefore, we can conclude that:
\[ \text{det}(A) \cdot \text{det}(B) \cdot \text{det}(C) = 0 \]

4. **Implication of the Determinant**:
The product of the determinants being zero implies that at least one of the determinants must be zero. This means:
\[ \text{det}(A) = 0 \quad \text{or} \quad \text{det}(B) = 0 \quad \text{or} \quad \text{det}(C) = 0 \]
Since A, B, and C are non-zero matrices, this means at least one of the matrices must be singular (not invertible).

5. **Evaluating the Statements**:
Now we need to evaluate the given statements:
- Statement 1: \( A = 0 \)
- Statement 2: \( B = 0 \)
- Statement 3: \( C = 0 \)
- Statement 4: None of these

Since we have established that at least one of the determinants must be zero, but we do not know which one, we cannot conclude that any specific matrix is zero. Therefore, the correct answer is that none of the statements can be definitively true.

### Conclusion:
The correct statement is: **None of these** (Option 4).

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