GATE 2025 ME – Question 11

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors · 1 mark · Multiple choice

Let **A** and **B** be real symmetric matrices of same size. Which one of the following options is correct?

$\mathbf{A}^T = \mathbf{A}^{-1}$
$\mathbf{AB} = \mathbf{BA}$
$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$
$\mathbf{A} = \mathbf{A}^{-1}$

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Correct answer: (C) $(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$

Explanation

The identity $(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$ holds for any two matrices. Symmetric matrices need not be orthogonal ($\mathbf{A}^T = \mathbf{A}^{-1}$), need not commute, and need not be their own inverse.