GATE DA 2025 | Question 52

MSQ · 2 marks

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

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

Answer

(A), (D)

Explanation

Write the condition as \(x^{\top}(A^{\top}A - I)x = 0\) for every \(x\). The matrix \(A^{\top}A - I\) is symmetric, and a symmetric matrix whose quadratic form is zero for every \(x\) is the zero matrix. So \(A^{\top}A = I\), and \(A\) is orthogonal: A holds.

  • D holds: \(A^{\top}A = I\) means \(A^{-1} = A^{\top}\) exists, so \(A\) has full rank.
  • B fails: \(-I\), or any rotation, also preserves lengths.
  • C fails: the rotation by 90\(^{\circ}\), \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\), is orthogonal, but its eigenvalues are \(\pm i\). An orthogonal matrix only guarantees eigenvalues of modulus 1.

Answer: A, D