Consider three orthogonal matrices A, B, and C and define X := AB and Y = ABC. Which of the following are true for all A, B, and C?


Choose the best option(s)

The inverse of X is not orthogonal

The matrix X + Y is orthogonal

The matrix X - Y is orthogonal

The matrix X is orthogonal

The matrix Y is orthogonal

Given the problem, let's analyze each option considering the properties of orthogonal matrices.

### Properties of Orthogonal Matrices:
1. A matrix \( A \) is orthogonal if \( A^T A = I \), where \( A^T \) is the transpose of \( A \) and \( I \) is the identity matrix.
2. The inverse of an orthogonal matrix \( A \) is also orthogonal, and it is given by \( A^{-1} = A^T \).
3. The product of two orthogonal matrices \( A \) and \( B \) is also orthogonal, i.e., if \( A \) and \( B \) are orthogonal, then \( AB \) is orthogonal.

### Let’s analyze each statement:

1. **The inverse of \( X \) is not orthogonal:**
   - Since \( A \) and \( B \) are orthogonal matrices, \( X = AB \) is also orthogonal. The inverse of an orthogonal matrix is also orthogonal. Therefore, this statement is **false**.

2. **The matrix \( X + Y \) is orthogonal:**
   - \( X \) and \( Y \) are both products of orthogonal matrices, but \( X + Y \) is not necessarily orthogonal. Orthogonality does not generally hold under addition. Therefore, this statement is **false**.

3. **The matrix \( X - Y \) is orthogonal:**
   - Similar to the case above, \( X - Y \) is not necessarily orthogonal, as orthogonality does not generally hold under subtraction. Therefore, this statement is **false**.

4. **The matrix \( X \) is orthogonal:**
   - Since \( X = AB \) and both \( A \) and \( B \) are orthogonal, \( X \) is also orthogonal. Therefore, this statement is **true**.

5. **The matrix \( Y \) is orthogonal:**
   - \( Y = ABC \) where \( A \), \( B \), and \( C \) are orthogonal matrices. The product of orthogonal matrices is also orthogonal, so \( Y \) is orthogonal. Therefore, this statement is **true**.

### Conclusion:
The correct statements are:
- **The matrix \( X \) is orthogonal.**
- **The matrix \( Y \) is orthogonal.**

These are the true statements for all orthogonal matrices \( A \), \( B \), and \( C \).

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Do you want more details or have any questions?

Here are 5 related questions to consider:

1. How does the determinant of an orthogonal matrix affect its properties?
2. What are the eigenvalues of an orthogonal matrix?
3. Can the sum of two orthogonal matrices ever be orthogonal?
4. How does the concept of orthogonality apply to vectors compared to matrices?
5. What are some applications of orthogonal matrices in real-world problems?

**Tip:** Remember that orthogonal matrices preserve the dot product and are often used in transformations that maintain lengths and angles, such as rotations.

Explore the properties of orthogonal matrices with examples. Learn why the matrices X = AB and Y = ABC behave as they do under matrix operations. Discover when matrices X and Y are orthogonal and when they are not.

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