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GATE 2026 AE – Question 22

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 1 mark · Multiple select

An $n \times n$ square matrix $A$ satisfies $A^T = A^{-1}$. The determinant of this matrix may take which of the following value(s)?

  1. $+1$
  2. $-1$
  3. $n$
  4. $0$

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Show answer and explanation

Correct answer: (A) $+1$; (B) $-1$

Explanation

A matrix with $A^T = A^{-1}$ is orthogonal. Then $\det(A)\det(A^T) = \det(A)^2 = 1$, so $\det A = \pm 1$.