A and B are square matrices of order `3xx3` , A is an orthogonal matrix and B is a skew symmetric matrix. Which of the following statement is not true

Options

A. (a) `|AB|=1`
B. (b) `|AB|=0`
C. (c) `|AB|=-1`
D. (d) none of these

Answer

To solve the problem, we need to analyze the properties of the matrices A and B based on the information given:

1. **Understanding Orthogonal Matrix (A)**:
- A matrix A is orthogonal if \( A A^T = I \), where \( A^T \) is the transpose of A and I is the identity matrix.
- The determinant of an orthogonal matrix is either +1 or -1. Thus, we can write:
\[ \text{det}(A) = \pm 1 \]

2. **Understanding Skew-Symmetric Matrix (B)**:
- A matrix B is skew-symmetric if \( B^T = -B \).
- For any skew-symmetric matrix of odd order (like 3x3), the determinant is always zero. Thus:
\[ \text{det}(B) = 0 \]

3. **Finding the Determinant of the Product (AB)**:
- The determinant of the product of two matrices is the product of their determinants:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
- Substituting the known values:
\[ \text{det}(AB) = (\pm 1) \cdot 0 = 0 \]

4. **Analyzing Statements**:
- Now, we need to evaluate the statements provided in the question to identify which one is not true based on our findings:
- Since we know that \( \text{det}(A) = \pm 1 \) and \( \text{det}(B) = 0 \), we conclude that \( \text{det}(AB) = 0 \).
- Any statement claiming that \( \text{det}(AB) \neq 0 \) would be false.

5. **Conclusion**:
- After evaluating the properties of both matrices and their determinants, we can conclude which statement is not true.

### Summary of the Solution:
- The determinant of an orthogonal matrix \( A \) is \( \pm 1 \).
- The determinant of a skew-symmetric matrix \( B \) of order 3 is \( 0 \).
- Therefore, the determinant of the product \( AB \) is \( 0 \).