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GATE 2024 MT – Question 28

Engineering Mathematics · Linear algebra and matrices · 1 mark · Multiple select

Which statements are correct for a real square matrix $A$? Here $A^T$ denotes transpose and $A^{-1}$ inverse.

  1. $A$ is symmetric if $A^T=-A$.
  2. $A$ is skew-symmetric if $A^T=-A$.
  3. If $A$ is orthogonal, then $A^T=A^{-1}$.
  4. If $A$ is orthogonal, its determinant is zero.

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Correct answer: (B) $A$ is skew-symmetric if $A^T=-A$.; (C) If $A$ is orthogonal, then $A^T=A^{-1}$.

Explanation

Skew symmetry means Aᵀ=−A, so B is true; ordinary symmetry instead requires Aᵀ=A.

Orthogonality gives AᵀA=I, hence Aᵀ=A⁻¹, making C true.

Its determinant is ±1, not zero.

Therefore **B,C**.