Question

If A and B be orthogonal matrices of same size, then

• A. \(|A| = \pm 1\)
• B. \(A^T = -A^{-1}\)
• C. \(|A| = 0\)
• D. AB is not orthogonal

Hint:

The determinant of any orthogonal matrix is always \( \pm 1 \).
- If \( \det(A) = 1 \), the transformation is a pure rotation.
- If \( \det(A) = -1 \), the transformation involves a reflection.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the definitions and mathematical properties of orthogonal matrices in linear algebra.

Step 2: Detailed Explanation:
Let us analyze the definition of an orthogonal matrix:
- A square matrix \( A \) is said to be orthogonal if its transpose is equal to its inverse:
\[ A^T = A^{-1} \] Multiplying both sides of the equation by \( A \) on the right:
\[ A A^T = A A^{-1} \implies A A^T = I \] where \( I \) is the identity matrix of the same size.
- Let us find the determinant properties of this orthogonal matrix:
Take the determinant on both sides of the orthogonal equation:
\[ \det(A A^T) = \det(I) \] We know that the determinant of the identity matrix is 1 (\( \det(I) = 1 \)), and the determinant of a product is the product of their determinants:
\[ \det(A) \cdot \det(A^T) = 1 \] A well-known property of determinants is that the determinant of a matrix is equal to the determinant of its transpose (\( \det(A^T) = \det(A) \)). Substitute this into the equation:
\[ \det(A) \cdot \det(A) = 1 \implies [\det(A)]^2 = 1 \] Taking the square root on both sides:
\[ \det(A) = \pm 1 \quad \text{or} \quad |A| = \pm 1 \] This proves that the determinant of any orthogonal matrix is always \( +1 \) or \( -1 \). Thus, Option (A) is correct.
Let us evaluate why other options are incorrect:
- Option (B): The definition requires \( A^T = A^{-1} \), not \( -A^{-1} \).
- Option (C): Since the determinant is \( \pm 1 \), it can never be 0 (orthogonal matrices are always non-singular/invertible).
- Option (D): If \( A \) and \( B \) are orthogonal, their product \( AB \) is also orthogonal:
\[ (AB)(AB)^T = (AB)(B^T A^T) = A(B B^T)A^T = A(I)A^T = A A^T = I \] Thus, \( AB \) is orthogonal. Therefore, this option is false.

Step 3: Final Answer:
The statement \( |A| = \pm 1 \) is always true.
Therefore, the correct choice is Option (A).