If `A and B` are two orthogonal matrices of order n and `det (A) + det (B) = 0;` then which of the following must be correct ?

Options

A. `det(A+B)=det(A)+det(B)`
B. `det(A+B)=0`
C. A and B both are singular matrices
D. `A+B=0`

Answer

To solve the problem, we need to analyze the properties of orthogonal matrices and their determinants. Let's break down the steps systematically.

### Step-by-Step Solution:

1. **Understanding Orthogonal Matrices**:
- A matrix \( A \) is orthogonal if \( A^T A = I \) (where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix).
- The determinant of an orthogonal matrix \( A \) satisfies \( \det(A) = \pm 1 \).

2. **Given Condition**:
- We are given that \( \det(A) + \det(B) = 0 \).
- This implies that \( \det(B) = -\det(A) \).

3. **Possible Values of Determinants**:
- Since both \( A \) and \( B \) are orthogonal, we have \( \det(A) = \pm 1 \) and \( \det(B) = \pm 1 \).
- If \( \det(A) = 1 \), then \( \det(B) = -1 \).
- If \( \det(A) = -1 \), then \( \det(B) = 1 \).

4. **Conclusion About Determinants**:
- From the above, we conclude that one of the matrices has a determinant of 1 and the other has a determinant of -1.

5. **Analyzing the Options**:
- Since neither \( \det(A) \) nor \( \det(B) \) can be zero (as they are both orthogonal matrices), we can conclude that neither \( A \) nor \( B \) is a singular matrix.
- Therefore, both matrices are non-singular.

6. **Final Conclusion**:
- The correct conclusion is that both \( A \) and \( B \) are non-singular matrices, meaning their determinants are non-zero.