If $P=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $Q=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ then $Q^T{P}^T$ is

**Matrix Transposition $Q^T{P}^T$ Calculation**

Given the matrices $P=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $Q=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, the task is to calculate the product of their transposes, $Q^T{P}^T$. This requires finding the transpose of each matrix individually and then multiplying them.

**Step 1: $P^T$ Calculation**

The transpose of matrix $P$, denoted as $P^T$, is found by switching the rows and columns of $P$.

$P^T={\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^T=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

**Step 2: $Q^T$ Calculation**

Similarly, the transpose of matrix $Q$, denoted as $Q^T$, is found by switching the rows and columns of $Q$. Note that $Q$ is a symmetric matrix, so its transpose is itself.

$Q^T={\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]}^T=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

**Step 3: $Q^T{P}^T$ Product**

Now, multiply the transpose matrices $Q^T$ and $P^T$ in that order.

$Q^T{P}^T=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

Performing the matrix multiplication:

$Q^T{P}^T=\left[\begin{array}{cc}(0\times 1)+(1\times 2)& (0\times 3)+(1\times 4)\\ (1\times 1)+(0\times 2)& (1\times 3)+(0\times 4)\end{array}\right]$

$Q^T{P}^T=\left[\begin{array}{cc}0+2& 0+4\\ 1+0& 3+0\end{array}\right]$

$Q^T{P}^T=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$

**Final $Q^T{P}^T$ Matrix**

The calculated matrix for $Q^T{P}^T$ is $\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$.