Question

An \( n \times n \) matrix \( A \) with real entries satisfies the property: \[ \|Ax\|^2 = \|x\|^2, \quad \text{for all } x \in \mathbb{R}^n, \] where \( \| \cdot \| \) denotes the Euclidean norm. Which of the following statements is/are ALWAYS correct?

• A. \( A \) must be orthogonal
• B. \( A = I \), where \( I \) denotes the identity matrix, is the only solution
• C. The eigenvalues of \( A \) are either +1 or -1
• D. \( A \) has full rank

Hint:

For a matrix to be orthogonal, it must satisfy \( A^T A = I \), meaning its rows and columns are orthonormal vectors. Orthogonal matrices always have full rank.

Answer 1

The Correct Option is A, D

Given that for all \( x \in \mathbb{R}^n \),
\[ \|Ax\|^2 = \|x\|^2, \] we derive:
\[ (Ax)^T (Ax) = x^T x \quad \Rightarrow \quad x^T A^T A x = x^T x. \] This implies \( A^T A = I \), so \( A \) is an orthogonal matrix.

Since \( A \) is orthogonal, it must have full rank.
The eigenvalues of orthogonal matrices lie on the unit circle, so they are not necessarily just +1 or -1.

Therefore, Option (A) is correct: \( A \) must be orthogonal, and Option (D) is correct: \( A \) has full rank.

Option (B) is incorrect because \( A \) can be any orthogonal matrix, not just the identity matrix, and Option (C) is incorrect because the eigenvalues of \( A \) are not restricted to +1 or -1.