The GATE Grind

GATE 2024 CE (CE2) – Question 12

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.
P: $(\mathbf{A} + \mathbf{B})^T = \mathbf{A}^T + \mathbf{B}^T$
Q: $(\mathbf{AB})^T = \mathbf{A}^T\mathbf{B}^T$

Which one of the following options is CORRECT?

  1. P is TRUE and Q is FALSE
  2. Both P and Q are TRUE
  3. P is FALSE and Q is TRUE
  4. Both P and Q are FALSE

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Show answer and explanation

Correct answer: (A) P is TRUE and Q is FALSE

Explanation

The transpose of a sum is the sum of the transposes, so P is true. The transpose of a product reverses the order, $(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$, which differs from $\mathbf{A}^T\mathbf{B}^T$ in general, so Q is false.