GATE 2021 CE (CE1) – Question 12

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

If $P=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $Q=\begin{bmatrix}0&1\\1&0\end{bmatrix}$, then $Q^TP^T$ is

$\begin{bmatrix}1&2\\3&4\end{bmatrix}$
$\begin{bmatrix}1&3\\2&4\end{bmatrix}$
$\begin{bmatrix}2&1\\4&3\end{bmatrix}$
$\begin{bmatrix}2&4\\1&3\end{bmatrix}$

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Correct answer: (D) $\begin{bmatrix}2&4\\1&3\end{bmatrix}$

Explanation

$P^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}$ and $Q^T=Q$. Left multiplication by Q interchanges the rows, giving $\begin{bmatrix}2&4\\1&3\end{bmatrix}$. **Answer: D.** The transpose reverses the multiplication order: $Q^TP^T=(PQ)^T$.